" 

2  7  1997 


HARPER'S  SCIENTIFIC  MEMOIRS 

EDITED  BY 

J.  S.  AMES,  PH.D. 

PROFESSOR    OF   PHYSICS    IN    JOHNS   HOPKINS    UNIVERSITY 


VI. 
THE  SECOND  LAW  OF  THERMODYNAMICS 


SCIENCE  &  ENGINEERING 
LIBRARY 

AUG  2  7  1997 

PHYSICS  COLLECTION 
UCLA 


THE    SECOND    LAW    OF 
THERMODYNAMICS 


MEMOIRS  BY  CARNOT,  CLAUSIUS 
AND  THOMSON 


TRANSLATED  AND  EDITED 

BY  W.  F.  MAGIE,  FH  D. 

PROFESSOR   OF   PHYSICS    IN    PRINCKTON    UNIVERSITY 


NEW  YORK  AND  LONDON 

HARPER    &    BROTHERS    PUBLISHERS 
1899 


HARPER'S   SCIENTIFIC   MEMOIRS. 

XDITKO    BT 

J.   S.  AMES,  PH.D., 
i-Boruwok  op  riiTsioa  IN  JOHNS  HOPKINS  VNIVKEBITY. 

KOW  READY: 
T1IK    KKKK    KXI'ANSION   OF  OASES.      Memoir* 

by  Gtty- I.iii»i«iic,  Joule,  and  Joule   unit  Thommin. 

Editor,    Priif.  .1.  S.   AMK*,   Ph.D.,  John*    Hopkins 

Unlverpity.    TAcenla. 
PUISMATIC     AND     DIFFRACTION     SPKrTKA 

Memoir*  by  J.wfph  von  Frminhof.-r.     Editor,  Prof. 

J.   S.    AMIOS    Ph.D.,   Johus    Hopkins    Univ.TMty. 

«0  cent*. 
UiiN'KJKN    KAYS.    Memoir*  by   RfiiitKeii,  Stoke*, 

:u,il    .1.   .!.     ThoMi*.!..     K.!it..,.    |'i,,i.    Qn 

HAKKKK,  Uiiiveifity  of  Punnoylvuiilu.     C<i  reuta. 
T1IK    MODI-UN    -HIKoliV    «.F    S.>IIT|«,N.      \|, 

molraby  Pf.-fT.-r.  Vnu't  l|.,fr.  Anl.rniu-.  an.l  KaoulU 

K«liii.r,  Dr.  II.  C.  JON«J>,  Joluut  Hopkliia  Ulilversily. 

TUB  LAWS  OF  ({ASKS.  Mem»lra  by  Boyle  nod 
AIIIH^.H.  K.lilor,  I'mU'AKi.  HAKI m  Hn.WnfnivprFily. 

TIIK   >K< -nM)   LAW   UK  -i  HI:I:M<>I>YN.\MI<  > 
tnot.CIawlM.udTlKMMaa  Editor, 
Prof.  W.  F.  MAUIK,  Princeton  t'lilvon-liy. 

/V   1-KKPA  RATIOS: 

TIIK  Ki  M.\MI:M  \l  LAW!  Of  i-:i  MTM 
I.YTlr  (ONDl't  'II  l.y  Fanulay, 

liittorf,  m..l    Kohln.'ii..:h.      K.lii..r.   D'.   II     M 
WIN,  Mni'i.iichilKolti.  IllMlllltV  oriVchllolotry. 

TIIK    KK !••(•: i   is    I.K   A    MAGNETIC    FIELD   ON 
I!AIU\'II"V      M.-nioin.    l.y    K.ir»c),iv,   Krrr,  mid 
itur,  Dr.  K  P.  LKWIR,  Unlvcn<ity  of 
Cnlirornla. 

THK  WAVB-TIH.oHv  nK  I.K.IIT.  M.-molm  by 
H.iyirniF.Y. .nni'.  m«l  Kr.-n.-l  K.lit..r.  Prof.  HRNRT 
Citrw,  Norihwentern  L'lilverelty. 

NKWTnNX    M\v    (iK    (iHAVlTATliiN.       Bditor, 
\    S  MAOKKKXIIT.  Bryn  Mnwr  College 


SICW    YORK    AMI    LONDON: 

HARPER  A  BROTHERS,  PUBLISHERS. 


A.    I;,,.,MM:, 


Qc 


PREFACE 


AFTER  the  invention  of  the  steam-engine  in  its  present  form 
by  James  Watt,  the  attention  of  engineers  and  of  scientific 
men  was  directed  to  the  problem  of  its  further  improvement. 
With  this  end  in  view,  the  young  Sadi  Carnot,  in  1824,  pub- 
lished the  Reflexions  sur  la  Puissance  Motrice  du  Feu,  of  which 
this  translation  is  given  in  this  volume.  In*  this  really  great 
memoir,  Carnot  examined  the  relations  between  heat  and  the 
work  done  by  heat  used  in  an  ideal  engine,  and  by  reducing 
the  problem  to  its  simplest  form  and  avoiding  all  special  ques- 
tions relating  to  details,  he  succeeded  in  establishing  the  condi- 
tions upon  which  the  economical  working  of  all  heat-engines 
depends.  It  is  not  necessary  here  to  animadvert  upon  the  use 
made  by  Carnot  of  the  substantial  theory  of  heat,  and  the  con- 
sequent failure  of  the  proof  of  his  main  proposition  when  the 
true  nature  of  heat  was  appreciated.  It  is  sufficient  to  say 
that  though  the  proof  was  invalid,  the  proposition  remained 
true,  and  carried  with  it  the  truth  of  such  of  Carnot's  deduc- 
tions as  were  based  solely  upon  it. 

Carnot's  memoir  remained  for  a  long  time  unappreciated, 
and  it  was  not  until  use  was  made  of  it  by  William  Thomson 
(now  Lord  Kelvin),  in  1848,  to  establish  an  absolute  scale  of 
temperature,  that  the  merits  of  the  method  proposed  in  it  were 
recognized.  In  his  first  paper  on  this  subject  Thomson  re- 
tained the  substantial  theory  of  heat,  but  the  evidence  in  favor 
of  the  mechanical  theory  became  so  strong  that  he  soon  after 
adopted  the  new  view.  Applying  it  to  the  questions  treated 
by  Carnot,  he  found  that  Carnot's  proposition  could  no  longer 
be  proved  by  denying  the  possibility  of  "  the  perpetual  motion," 
and  was  led  to  lay  down  a  second  fundamental  principle  to  serve 
in  the  demonstration.  This  principle  is  now  called  the  Second 
Law  of  Thermodynamics.  A  part  of  the  memoir  in  which  this 


PREFACE 

principle  is  stated  :»ml  many  of  its  consequences  developed  is 
given  in  this  volume.     It  was  publislu-d  in  March.  is.M. 

In  the  previous  year  Clansius  published  a  discussion  of  the 
same  question  as  that  treated  by  Thomson,  in  which  he  lays 
down  a  principle  for  use  in  the  demonstration  of  Cairnot's  propo- 
sitton.  which,  while  not  the  same  in  form  as  Thomson's,  is  the 
same  in  content,  and  ranks  as  another  statement  of  the  Second 
Law  of  Thermodynamics.  \\\^  paper  is  also  given  in  this  vol- 
ume. While  not  so  powerful  or  so  inclusive  as  Thomson's,  it 
deserves  attention  for  the  clearness  and  simplicity  of  its  form. 
Clausing  followed  up  this  paper  by  others,  and  subsequently 
published  a  book  in  which  the  subject  of  Thermodynami--- 
Lriven  a  systematic  treatment,  and  in  which  he  introduced  and 
developed  the  important  function  called  by  him  the  entropy. 

The  science  of  Thermodynamics,  founded  by  the  labors  of 
these  three  illustrious  men,  has  led  to  the  most  important  de- 
velopments in  all  departments  of  physical  science.  It  lia- 
pointed  out  relations  among  the  properties  of  bodies  which 
could  scarcely  have  been  anticipated  in  any  other  way  :  it  ha- 
laid  the  foundation  for  the  Science  of  Chemical  Physics  ;  and. 
taken  in  connection  with  the  kinetic  theory  of  gases,  as  devel- 
oped by  Maxwell  and  Boltzmann,  it  has  furnished  a  funeral 
view  of  the  operations  of  the  universe  which  is  far  in  advance 
of  any  that  could  have  been  reached  by  purely  dynamical  rea- 
soning. 


GENERAL   CONTENTS 


PAGE 

Preface v 

Reflections  on  the  Motive  Power  of  Heat.     By  Sadi  Carnot 3 

Biographical  Sketch  of  Carnot 60 

Ou  the  Motive  Power  of  Heat,  and  on  the  Laws  which  can  be  De- 
duced from  it  for  the  Theory  of  Heat.     By  R.  Clausius 65 

Biographical  Sketch  of  Clausius 107 

The  Dynamical  Theory  of  Heat.     (Selected  Portions.)    By  William 

Thomson  (Lord  Kelvin) Ill 

Biographical  Sketch  of  Lord  Kelvin 147 

Bibliography 149 

INDEX..  ..  151 


REFLECTIONS   ON 
THE  MOTIVE  POWER  OF  HEAT 

BY 

SADI    CAENOT 

Paris,  1824 


CONTENTS 

PAG  I 

Ueat-enginet 8 

JbB  of  Temperature 7 

Reversible  Processes 11 

"Carnot's  Cycle  " 10 

Efficiency  a  Function  of  Limiting  Temperatures -2(\ 

Specific  Heats  of  Gate* 21 

Motive  Ptoeer  of  Air,  Steam,  Alcohol  Vapor 40 

Greatest  Efficiency 49 

I  ;,rinvt  Types  of  Machines 62 

Advantages  and  Disadvantages  of  Steam 55 


REFLECTIONS   ON 

THE  MOTIVE  POWER  OF  HEAT  AND 

ON    ENGINES    SUITABLE    FOR 

DEVELOPING  THIS  POWER 

BY 

SADI    CARNOT 

IT  is  well  known  that  heat  may  be  nsed  as  a  cause  of  motion, 
and  that  the  motive  power  which  may  be  obtained  from  it  is 
very  great.  The  steam-engine,  now  in  such  general  use,  is  a 
manifest  proof  of  this  fact. 

To  the  agency  of  heat  may  be  ascribed  those  vast  disturb- 
ances which  we  see  occurring  everywhere  on  the  earth  ;  the 
movements  of  the  atmosphere,  the  rising  of  mists,  the  fall  of 
rain  and  other  meteors,*  the  streams  of  water  which  channel 
the  surface  of  the  earth,  of  which  man  has  succeeded  in  util- 
izing only  a  small  part.  To  heat  are  due  also  volcanic  erup- 
tions and  earthquakes.  From  this  great  source  we  draw  the 
moving  force  necessary  for  our  use.  Nature,  by  supplying 
combustible  material  everywhere,  has  afforded  us  the  means  of 
generating  heat  and  the  motive  power  which  is  given  by  it,  at 
all  times  and  in  all  places,  and  the  steam-engine  has  made  it 
possible  to  develop  and  use  this  power. 

The  study  of  the  steam-engine  is  of  the  highest  interest, 
owing  to  its  importance,  its  constantly  increasing  use,  and  the 
great  changes  it  is  destined  to  make  in  the  civilized  world.  It 
has  already  developed  mines,  propelled  ships,  and  dredged 
rivers  and  harbors.  It  forges  iron,  saws  wood,  grinds  grain, 
spins  and  weaves  stuffs,  and  transports  the  heaviest  loads.  In 
the  future  it  will  most  probably  be  the  universal  motor,  and 

*  [Any  atmospJieric  phenomenon  was  formerly  called  a  meteor.] 


MKMuIKS    ON 

will  furnish  the  power  now  obtained  from  animals,  from  water- 
falls, and  from  air-currents.  Over  the  first  of  these  motors  it 
has  the  advantage  of  economy,  and  over  the  other  two  the  in- 
calculable advantage  that  it  can  be  used  everywhere  and  always. 
and  that  its  work  need  never  be  interrupted.  If  in  the  future 
the  steam-engine  is  so  perfected  as  to  render  it  less  costly  to 
construct  it  and  to  supply  it  with  fuel,  it  will  unite  all  desir- 
able qualities  and  will  promote  the  development  of  the  indus- 
trial arts  to  an  extent  which  it  is  difficult  to  foresee.  It  is,  in- 
deed, not  only  a  powerful  and  convenient  motor,  which  can  be 
set  up  or  transported  anywhere,  and  substituted  for  other 
motors  already  in  use,  but  it  lends  to  the  rapid  extension  of 
those  arts  in  which  it  is  used,  and  it  can  even  create  arts  hith- 
erto unknown. 

The  most  signal  service  which  has  been  rendered  to  England 
by  the  steam-engine  is  that  of  having  revived  the  working  of 
her  coal-mines,  which  had  languished  and  was  threatened  with 
extinction  on  account  of  the  increasing  ditlirulty  of  excavation 
and  extraction  of  the  coal.*  We  may  place  in  the  second  rank 
the  services  rendered  in  the  manufacture  of  iron,  as  much  by 
furnishing  an  abundant  supply  of  coal,  which  took  the  place  of 
wood  as  the  wood  began  to  be  exhan-te.l.  a-  l>y  the  powerful 
machines  of  all  kinds  tho  use  of  which  it  either  facilitated  or 
made  possible. 

Iron  and  fire,  as  every  one  knows,  are  the  mainstays  of  the 
mechanical  arts.  Perhaps  there  is  not  in  all  England  a  single 
industry  whose  existence  is  not  dependent  <>n  these  agents,  and 
which  does  not  use  them  extensively.  If  England  were  to-day 
to  lose  its  steam-engines  it  would  lose  also  its  coal  and  iron, 
and  this  loss  would  dry  up  all  its  sources  of  wealth  and  destroy 
its  prosperity ;  it  would  annihilate  this  colossal  power.  The 
destruction  of  its  navy,  which  it  considers  its  strongest  support, 
would  be,  perhaps,  less  fatal. 

The  safe  and  rapid  navigation  by  means  of  steamships  is  an 

•  One  may  wifely  say  that  the  mining  of  coal  has  increased  tenfold  since 

the  Jnv.-nti. f  tho  steam  engine.     Tin*  mining  of  copper,  <>f  tin.  mid  <>f 

iron  has  increased  almost  as  much.  The  effect  produced  half  n  century 
ago  in  the  mines  of  Knd.ind  i>  imw  IH-III^  r<  \« -uted  in  the  irnld  and  -liver 
mines  of  the  New  W<H|,|,  the  working  of  which  was  steadily  declining. 
principally  on  account  of  the  insulli. •!•  -ncy  of  the  motors  used  for  the  ex- 
cavation and  extraction  of  the  minerals. 
4 


THE    SECOND    LAW    OF    THERMODYNAMICS 

entirely  new  art  due  to  the  steam-engine.  This  art  has  already 
made  possible  the  establishment  of  prompt  and  regular  com- 
munication on  the  arms  of  the  sea,  and  on  the  great  rivers  of 
the  old  and  new  continents.  By  means  of  the  steam-engine 
regions  still  savage  have  been  traversed  which  but  a  short  time 
ago  could  hardly  have  been  penetrated.  The  products  of  civ- 
ilization have  been  taken  to  all  parts  of  the  earth,  which  they 
would  otherwise  not  have  reached  for  many  years.  The  navi- 
gation due  to  the  steam-engine  has  in  a  measure  drawn  to- 
gether the  most  distant  nations.  It  tends  to  unite  the  peoples 
of  the  earth  as  if  they  all  lived  in  the  same  country.  In  fact, 
to  diminish  the  duration,  the  fatigue,  the  uncertainty  and  dan- 
ger of  voyages  is  to  lessen  their  length.* 

The  discovery  of  the  steam-engine,  like  most  human  inven- 
tions, owes  its  birth  to  crude  attempts  which  have  been  attrib- 
uted to  various  persons  and  of  which  the  real  author  is  not 
known.  The  principal  discovery  consists  indeed  less  in  these 
first  trials  than  in  the  successive  improvements  which  have 
brought  it  to  its  present  perfection.  There  is  almost  as  great 
a  difference  between  the  first  structures  where  expansive  force 
was  developed  and  the  actual  steam-engine  as  there  is  between 
the  first  raft  ever  constructed  and  a  man-of-war. 

If  the  honor  of  a  discovery  belongs  to  the  nation  where  it 
acquired  all  its  development  and  improvement,  this  honor  can- 
not in  this  case  be  withheld  from  England  :  Savery,  Newcomen, 
Smeaton,  the  celebrated  Watt,  Woolf,  Trevithick,  and  other 
English  engineers,  are  the  real  inventors  of  the  steam-engine. 
At  their  hands  it  received  each  successive  improvement.  It  is 
natural  that  an  invention  should  be  made,  improved,  and  per- 
fected where  the  need  of  it  is  most  strongly  felt. 

In  spite  of  labor  of  all  sorts  expended  on  the  steam-engine, 
and  in  spite  of  the  perfection  to  which  it  has  been  brought, 
its  theory  is  very  little  advanced,  and  the  attempts  to  bet- 
ter this  state  of  affairs  have  thus  far  been  directed  almost  at 
random. 

The  question  has  often  been  raised  whether  the  motive  power 

*  We  speak  of  diminishing  the  danger  of  voyages  ;  in  fact,  though  the 
use  of  the  steam-engine  in  ships  is  attended  with  some  dangers,  these  are 
al ways  exaggerated  and  are  compensated  for  by  the  ability  of  ships  to  keep 
a  definite  course,  and  to  resist  winds  which  would  otherwise  drive  the  ves- 
sel on  the  coast,  or  on  shoals  or  reefs. 
5 


MEMOIRS    OX 

of  heat  is  limited  or  not  ;*  whether  there  is  a  limit  to  the  pos- 
sible improvements  of  the  steam-engine  which,  in  the  nature  of 
the  case,  cannot  be  passed  by  any  means ;  or  if,  on  the  other 
hand,  these  improvements  are  capable  of  indefinite  extension. 
Inventors  have  tried  for  a  long  time,  and  are  still  trying,  to 
find  whether  there  is  not  a  more  efficient  agent  than  water  Ky 
which  to  develop  the  motive  power  of  heat;  whether,  for  e\- 
ample,  atmospheric  air  does  not  offer  great  advantages  in  this 
respect.  We  propose  to  submit  these  questions  to  a  critical 
examination. 

The  phenomenon  of  the  production  of  motion  by  heat  hus 
not  been  considered  in  a  sufficiently  general  way.  It  has  l»een 
treated  only  in  connection  with  machines  whoso  nature  ami 
mode  of  action  do  not  admit  of  a  full  investigation  of  it.  In 
such  machines  the  phenomenon  is,  in  a  measure,  imperfect  and 
incomplete  ;  it  thus  becomes  difficult  to  recognize  its  principles 
and  study  its  laws.  To  examine  the  principle  of  t lie  production 
of  motion  by  heat  in  all  its  generality,  it  must  be  conceive.!  in- 
dependently of  any  mechanism  or  of  any  particular  agent  ;  it 
is  necessary  to  establish  proofs  applicable  not  only  to  steam- 
engines  f  but  to  all  other  heat-engines,  irrespective  of  the  work- 
ing substance  and  the  manner  in  which  it  acts. 

The  machines  which  are  not  worked  by  heat — for  instance, 
those  worked  by  men  or  animals,  by  water-falls,  or  by  air  cur- 
rents— can  be  studied  to  their  last  details  by  the  principles  of 
mechanics.  All  possible  cases  may  be  anticipated,  all  imagi- 
nable actions  are  subject  to  general  principles  already  well  es- 
tahlished  ;md  applicable  in  all  circumstances.  The  theory  of 
snch  machines  is  complete.  Such  a  theory  is  evidently  lacking 
for  heat-engines.  We  shall  never  possess  it  until  the  laws  of 
physics  are  so  extended  and  generalixed  as  to  make  known  in 
advance  all  the  effects  of  heat  acting  in  a  definite  \\-.\\  on  any 
body  whatsoever. 

*  The  expression  motive  power  here  signifies  the  useful  cfTcci  that  a 
motor  in  capable  of  producing.  This  effect  may  always  he  nnnsiin.1  in 
terms  of  Ibe  elevation  of  a  weight  through  a  certain  distance  ;  it  is  u>.  .1- 
UP'. I.  as  it  well  known,  by  the  product  of  the  weight  ami  the  In  ijiii  to 
which  it  is  raised. 

f  We  distinguish  hen-  between  the  stcnm-cnginc  and  the  heat-engine  in 
general. which  can  be  worked  by  nny  agent,  and  uot  by  water  vapor  only, 
to  realize  the  motive  power  of  heat. 
6 


THE    SECOND    LAW    OF    THERMODYNAMICS 

We  shall  take  for  granted  in  what  follows  a  knowledge,  at 
least  a  superficial  one,  of  the  various  parts  which  compose  an 
ordinary  steam-engine.  "We  think  it  unnecessary  to  describe 
the  fire-box,  the  boiler,  the  steam -chest,  the  piston,  the  con- 
denser, etc. 

The  production  of  motion  in  the  steam-engine  is  always  ac- 
companied by  a  circumstance  which  we  should  particularly  no- 
tice. This  circumstance  is  the  re-establishment  of  equilibrium 
in  the  caloric* — that  is,  its  passage  from  one  body  where  the 
temperature  is  more  or  less  elevated  to  another  where  it  is 
lower.  What  happens,  in  fact,  in  a  steam-engine  at  work? 
The  caloric  developed  in  the  fire-box  as  an  effect  of  combus- 
tion passes  through  the  wall  of  the  boiler  and  produces  steam, 
incorporating  itself  with  the  steam  in  some  way.  This  steam, 
carrying  the  caloric  with  it,  transports  it  first  into  the  cylinder, 
where  it  fulfils  some  function,  and  thence  into  the  condenser, 
where  the  steam  is  precipitated  by  coming  in  contact  with  cold 
water.  As  a  last  result  the  cold  water  in  the  condenser  receives 
the  caloric  developed  by  combustion.  It  is  warmed  by  means 
of  the  steam,  as  if  it  had  been  placed  directly  on  the  fire-box. 
The  steam  is  here  only  a  means  of  transporting  caloric  ;  it  thus 
fulfils  the  same  office  as  in  the  heating  of  baths  by  steam, 
with  the  exception  that  in  the  case  in  hand  its  motion  is  ren- 
dered useful. 

We  can  easily  perceive,  in  the  operation  which  we  have  just 
described,  the  re- establishment  of  equilibrium  in  the  caloric 
and  its  passage  from  a  hotter  to  a  colder  body.  The  first  of 
these  bodies  is  the  heated  air  of  the  fire-box ;  the  second,  the 
water  of  condensation.  The  re-establishment  of  equilibrium  of 
the  caloric  is  accomplished  between  them  —  if  not  complete- 
ly, at  least  in  part ;  for,  on  the  one  hand,  the  heated  air  after 
having  done  its  work  escapes  through  the  smoke-stack  at  a 
much  lower  temperature  than  that  which  it  had  acquired  by 
the  combustion  ;  and,  on  the  other  hand,  the  water  of  the  con- 
denser, after  having  precipitated  the  steam,  leaves  the  engine 
with  a  higher  temperature  than  that  which  it  had  when  it 
entered. 

The  production  of  motive   power  in  the  steam-engine  is 


*  [Caloric  u  lieat  considered  as  an  indestructible  substance.     T/ie  word  is 
used  by  Carnot  interchangeably  with  fen,  fire,  or  heat.] 

7 


MKMOIRS    ON 

therefore  not  due  to  a  real  consumption  of  the  caloric,  but  t«  //>• 
transfer  from  a  hotter  /<>  a  n>1tli>r  fault/  —  that  is  to  say,  to  tin-  iv- 
establishment  of  its  equilibrium,  which  is  assumed  to  have  been 
destroyed  by  a  chemical  action  such  as  combustion,  or  by  some 
other  cause.  We  shall  soon  see  that  this  principle  is  applica- 
ble to  all  engines  operated  by  heat. 

According  to  this  principle,  to  obtain  motive  power  it  is  not 
enough  to  produce  heat  ;  it  is  also  necessary  to  provide  cold, 
without  which  the  heat  would  be  useless.  For  if  there  exist- 
ed only  bodies  as  warm  as  our  furnaces,  how  would  the  con- 
densation of  steam  be  possible,  and  where  could  it  be  sent  if  it 
were  once  produced?  It  cannot  be  replied  that  it  could  be 
ejected  into  the  atmosphere,  as  is  done  with  certain  engines,* 
since  the  atmosphere  would  not  receive  it.  In  the  actual  state 
of  things  the  atmosphere  acts  as  a  vast  condenser  for  the  steam. 
because  it  is  at  a  lower  temperature  ;  otherwise  it  would  soon 
be  saturated,  or,  rather,  would  be  saturated  in  advance.f 

Everywhere  where  there  is  a  difference  of  temperature,  and 
where  the  re-establishment  of  equilibrium  of  the  caloric  can  lie 
effected,  the  production  of  motive  power  is  possible.  Water 
vapor  is  one  agent  for  obtaining  this  power,  but  it  is  not  the 
only  one  ;  all  natural  bodies  can  be  applied  to  this  purpose,  for 
they  are  all  susceptible  to  changes  of  volume,  to  successive 
contractions  and  dilatations  effected  by  alternations  of  heat  and 
cold  ;  they  are  all  capable,  by  this  change  of  volume,  of  over- 
coming resistances  and  thus  of  developing  motive  power.  A 


*  Some  high  -pressure  engines  eject  vapor  into  the  atmosphere 
of  condensing  it.     They  are  used  mostly  in  places  where  it  is  difficult  to 
procure  a  current  of  cold  water  sutn.-i.-ni  t»  ctTcct  condensation. 

t  The  existence  of  water  in  a  liquid  state,  which  is  here  necessarily  as- 
sumed, since  without  it  the  steam  engine  could  not  be  supplied,  presup- 
poses the  existence  of  a  pressure  capable  of  preventing  it  from  evaporating, 
and  consequently  of  a  pressure  equal  to  or  greater  than  the  tension  of  the 
vapor  at  the  temperature  of  the  water.  If  such  a  pressure  were  not  ex- 
erted by  the  atmosphere  a  quantity  of  water  vapor  would  instantly  be  pro- 
duced sufficient  to  exert  this  pressure  on  itself,  and  this  pressure  must 
always  be  overcome  in  ejecting  the  steam  of  the  engine  into  the  new  at- 
mosphere. This  is  evidently  equivalent  to  overcoming  the  ton-inn  whi.-ii 
is  exerted  by  the  vapor  after  it  has  been  condensed  by  the  ordinary  means. 

If  a  very  high  temperature  were  to  prevail  at  the  surface  of  the  earth,  as 
it  almost  certainly  does  in  its  interior,  all  the  water  of  the  oceans  would 
exist  in  the  form  of  vapor  in  the  atmosphere,  and  there  would  be  no  water 
in  a  liquid  state. 

8 


THE    SECOND    LAW    OF    THERMODYNAMICS 

solid  body,  such  as  a  metallic  bar,  when  alternately  heated  and 
cooled,  increases  and  diminishes  in  length  and  can  move  bod- 
ies fixed  at  its  extremities.  A  liquid,  alternately  heated  and 
cooled,  increases  and  diminishes  in  volume  and  can  overcome 
obstacles  more  or  less  great  opposed  to  its  expansion.  An 
aeriform  fluid  undergoes  considerable  changes  of  volume  with 
changes  of  temperature  ;  if  it  is  enclosed  in  an  envelope  capa- 
ble of  enlargement,  such  as  a  cylinder  furnished  with  a  piston,  it 
will  produce  movements  of  great  extent.  The  vapors  of  all  bod- 
ies which  are  capable  of  evaporation,  such  as  alcohol,  mercury, 
sulphur,  etc.,  can  perform  the  same  function  as  water  vapor. 
This,  when  alternately  heated  and  cooled,  will  produce  motive 
power  in  the  same  way  as  permanent  gases,  without  returning  to 
the  liquid  state.  Most  of  these  means  have  been  proposed,  several 
have  been  even  tried,  though,  thus  far,  without  much  success. 

We  have  explained  that  the  motive  power  in  the  steam-engine 
is  due  to  a  re-establishment  of  equilibrium  in  the  caloric  ;  this 
statement  holds  not  only  for  steam-engines  but  also  for  all  heat- 
engines — that  is  to  say,  for  all  engines  in  which  caloric  is  the 
motor.  Heat  evidently  can  be  a  cause  of  motion  only  through 
the  changes  of  volume  or  of  form  to  which  it  subjects  the  body  ; 
those  changes  cannot  occur  at  a  constant  temperature,  but  are 
due  to  alternations  of  heat  and  cold  ;  thus  to  heat  any  sub- 
stance it  is  necessary  to  have  a  body  warmer  than  it,  and  to 
cool  it,  one  cooler  than  it.  We  must  take  caloric  from  the 
first  of  these  bodies  and  transfer  it  to  the  second  by  means  of 
the  intermediate  body,  which  transfer  re-establishes,  or,  at  least, 
tends  to  re-establish,  equilibrium  of  the  caloric. 

At  this  point  we  naturally  raise  an  interesting  and  important 
question  :  Is  the  motive  power  of  heat  invariable  in  quantity, 
or  does  it  vary  with  the  agent  which  one  uses  to  obtain  it — 
that  is,  with  the  intermediate  body  chosen  as  the  subject  of  the 
action  of  heat  ? 

It  is  clear  that  the  question  thus  raised  supposes  given  a  cer- 
tain quantity  of  caloric*  and  a  certain  difference  of  temperature. 

*  It  is  unnecessary  to  explain  here  what  is  meant  by  a  quantity  of  ca- 
loric or  of  heat  (for  we  use  the  two  expressions  interchangeably),  or  to  de- 
scribe how  these  quantities  are  measured  by  the  calorimeter  ;  nor  shall  we 
explain  the  terms  latent  heat,  degree  of  temperature,  specific  heat,  etc. 
The  reader  should  be  familiar  with  these  expressions  from  his  study  of  the 
elementary  treatises  of  physics  or  chemistry. 
9 


M  HMO  IRS    OX 

For  example,  we  suppose  that  we  have  at  onr  disposal  a  body.  .  I . 
maintained  at  the  temperature  100  degrees,  and  another  Im.ly. 
B,  at  0  degrees,  and  inquire  what  quantity  of  motive  power 
will  be  produced  by  the  transfer  of  a  given  quantity  of  caloric — 
for  example,  of  so  much  as  is  necessary  to  melt  a  kilogram  of 
ice — from  the  first  of  these  bodies  to  the  second ;  we  inquire  if 
this  quantity  of  motive  power  is  necessarily  limited ;  if  it  varies 
with  the  substance  used  to  obtain  it;  if  water  vapor  offers  in 
this  respect  more  or  less  advantage  than  vapor  of  alcohol  or  of 
mercury,  than  a  permanent  gas  or  than  any  other  substance. 
We  shall  try  to  answer  these  questions  in  the  light  of  the  con- 
siderations already  advanced. 

We  have  previously  called  attention  to  the  fact,  which  is  self- 
evident,  or  at  least  becomes  so  if  we  take  into  consideration  the 
changes  of  volume  occasioned  by  heat,  that  wherrrrr  ///>/•>  {.*  a 
difference  of  temperature  tkt  production  <i/ni»fin-  jtmn'r  is  />ns*i/i/,: 
Conversely,  wherever  this  power  can  be  employed,  it  is  possible 
to  produce  a  difference  of  temperature  or  to  destroy  the  equili- 
brium of  the  caloric.  Percussion  and  friction  of  bodies  are 
means  of  raising  their  temperature  spontaneously*  to  a  higher 
degree  than  that  of  surrounding  bodies,  and  consequently  of 
destroying  that  equilibrium  in  the  caloric  which  had  previously 
existed.  It  is  an  experimental  fact  that  the  temperature  of 
gaseous  fluids  is  raised  by  compression  and  lowered  by  expan- 
sion. This  is  a  sure  method  of  changing  the  temperature  of 
bodies,  and  thus  of  destroying  the  equilibrium  of  the  caloric  in 
the  same  substance,  as  often  as  we  please.  Steam,  when  used 
in  a  reverse  way  from  that  in  which  it  is  used  in  the  steam- 
engine,  can  thus  be  considered  as  a  means  of  destroying  the 
equilibrium  of  the  caloric.  To  be  convinced  of  this,  it  is  only 
necessary  to  notice  attentively  the  way  in  which  motive  power 
is  developed  by  the  action  of  heat  on  water  vapor.  Let  us 
consider  two  bodies,  A  and  B,  each  maintained  ut  a  constant 
temperature,  that  of  A  being  higher  than  that  of  />' :  these  two 
bodies,  which  can  either  give  up  or  receive  heat  without  a 
change  of  temperature,  perform  the  funetions  of  two  indefi- 
nitely great  reservoirs  of  calorie.  We  will  call  the  first  body 
the  source  and  the  second  the  refrigerator. 

If  we  desire  to  produce  motive  power  by  the  transfer  of  a 

*  [  That  it,  without  the  communication  of  heat.] 
10 


THE    SECOND    LAW    OF    THERMODYNAMICS 

certain  quantity  of  heat  from  the  body  A  to  the  body  B  we 
may  proceed  in  the  following  way : 

1.  We  take  from  the  body  A  a  quantity  of  caloric  to  make 
steam — that  is,  we  cause  A  to  serve  as  the  fire-pot,  or  rather 
as  the  metal  of  the  boiler  in  an  ordinary  engine  ;  we  assume 
the  steam   produced  to   be  at  the  same  temperature   as   the 
body  A. 

2.  The  steam  is  received  into  an  envelope  capable  of  enlarge- 
ment, such  as  a  cylinder  furnished  with  a  piston.     We  then  in- 
crease the  volume  of  this  envelope,  and  consequently  also  the 
volume  of  the  steam.     The  temperature  of  the  steam  falls  when 
it  is  thus  rarefied,  as  is  the  case  with  all  elastic  fluids ;  let  us 
assume  that  the  rarefaction  is  carried  to  the  point  where  the 
temperature  becomes  precisely  that  of  the  body  B. 

3.  We  condense  the  steam  by  bringing  it  in  contact  with  B 
and  exerting  on  it  at  the  same  time  a  constant  pressure  until  it 
becomes  entirely  condensed.     The  body  B  here  performs  the 
function  of  the  injected  water  in  an  ordinary  engine,  with  the 
difference  that  it  condenses  the  steam  without  mixing  with  it 
and  without  changing  its  own  temperature.*     The  operations 
which  we  have  just  described  could  have  been  performed  in  a 
reverse  sense  and  order.     There  is  nothing  to  prevent  the  for- 

*  It  will  perhaps  excite  surprise  that  B,  being  at  the  same  temperature 
as  the  steam,  can  condense  it.  Without  doubt  this  is  not  rigorously  possi- 
ble, but  the  smallest  difference  in  temperature  will  determine  condensa- 
tion. This  remark  is  sufficient  to  establish  the  propriety  of  our  reasoning. 
In  the  same  way,  in  the  differential  calculus,  to  obtain  an  exact  result  it  is 
sufficient  to  be  able  to  conceive  of  the  quantities  neglected  as  capable  of 
being  indefinitely  diminished  relative  to  the  quantities  retained  in  the  equa- 
tion. 

The  body  B  condenses  the  steam  without  changing  its  own  temperature. 
We  have  assumed  that  this  body  is  maintained  at  a  constant  temperature. 
The  caloric  is  therefore  taken  from  it  as  fast  as  it  is  given  up  to  it  by  the 
steam.  An  example  of  such  a  body  is  furnished  by  the  metallic  walls  of 
the  condenser  when  the  vapor  is  condensed  in  it  by  means  of  cold  water 
applied  to  the  outside,  as  is  done  in  some  engines.  In  the  same  way  the 
water  of  a  reservoir  can  be  maintained  at  a  constant  level,  if  the  liquid  runs 
out  at  one  side  as  fast  as  it  comes  in  at  the  other. 

One  could  even  conceive  the  bodies  A  and  B  such  that  they  would  re- 
main of  themselves  at  a  constant  temperature  though  losing  or  gaining 
quantities  of  heat.  If,  for  example,  the  body  A  were  a  mass  of  vapor 
ready  to  condense  and  the  body  B  a  mass  of  ice  ready  to  melt,  these  bodies, 
as  is  well  known,  could  give  out  or  receive  caloric  without  changing  their 
temperature. 

11 


MEMOIRS    ON 

mat  ion  of  vapor  by  means  of  the  caloric  of  the  body  B,  and  its 
compression  from  the  temperature  of  11,  in  such  a  way  that  it 
acquires  the  temperature  of  the  body  A,  and  then  its  condensa- 
tion in  contact  with  A,  under  a  pressure  which  is  maintained 
constant  until  it  is  completely  liquefied. 

In  the  first  series  of  operations  there  is  at  the  same  time  a 
production  of  motive  power  and  a  transfer  of  caloric  from  the 
body  A  to  the  body  R ;  in  the  reverse  series  there  is  at  the 
same  time  an  expenditure  of  motive  power  and  a  return  of  the 
caloric  from  B  to  A.  Hut  if  in  each  case  the  sumo  quantity  of 
vapor  has  been  used,  if  there  is  no  loss  of  motive  power  or  of 
caloric,  the  quantity  of  motive  power  produced  in  the  first 
case  will  equal  the  quantity  expended  in  the  second,  and  the 
quantity  of  caloric  which  in  the  first  case  passed  from  A  to  B 
will  equal  the  quantity  which  in  the  second  case  returns  from 
B  to  A,  so  that  an  indefinite  number  of  such  alternating  oper- 
ations can  be  effected  without  the  production  of  motive  power 
or  the  transfer  of  caloric  from  one  body  to  the  other.  Now  if 
there  were  any  method  of  using  heat  preferable  to  that  whieh 
we  have  employed,  that  is  to  say,  if  it  were  possible  that  the 
caloric  should  produce,  by  any  process  whatever,  a  larger  quan- 
tity of  motive  power  than  that  produced  in  our  first  series  of 
operations,  it  would  be  possible,  by  diverting  a  portion  of  this 
power,  to  effect  a  return  of  caloric,  by  the  method  just  indi- 
cated, from  the  body  B  to  the  body  A — that  is,  from  the  refrig- 
erator to  the  source — and  thus  to  re-establish  things  in  their 
original  state,  and  to  put  them  in  position  to  recommence  an 
operation  exactly  similar  to  the  first  one,  and  so  on  :  there 
would  thus  result  not  only  the  perpetual  motion,  but  an  indef- 
inite creation  of  motive  power  without  consumption  of  caloric 
or  of  any  other  agent  whatsoever.  Such  a  creation  is  entirely 
contrary  to  the  ideas  now  accepted,  to  the  laws  of  mechanics 
and  of  sound  physics ;  it  is  inadmissible.*  We  may  hence  con- 

*  The  objection  will  perhaps  here  be  made  that  perpetual  motion  has 
only  been  demonstrated  lo  be  impossible  in  the  case  of  mechanical  actions, 
and  that  it  may  not  be  so  when  we  employ  the  agency  of  lira: 
iricity  ;  but  can  we  conceive  of  the  phenomena  of  heat  ami  <.f  «•!••<  triciiy 
a*  due  lo  any  other  cause  than  some  motion  of  bodies,  nml,  as  such.  slnmM 
they  not  be  subject  to  the  general  laws  of  mechanics?  15-  -i  !•  -.  <l  .  \\,-  not 
know  a  potttriori  that  all  the  attempts  made  to  produce  perpetual  motion 
by  any  means  whatever  have  been  fruitless ;  that  no  truly  perpetual  inutiou 


THE    SECOND    LAW    OF    THERMODYNAMICS 

elude  that  the  maximum  motive  power  resulting  from  the  use  of 
steam  is  also  the  maximum  motive  power  which  can  be  obtained 
by  any  other  means.  We  shall  soon  give  a  second  and  more 
rigorous  demonstration  of  this  law.  What  has  been  given 
should  only  be  regarded  as  a  sketch  (see  page  15). 

It  may  properly  be  asked,  in  connection  with  the  proposition 
just  stated,  what  is  the  meaning  of  the  word  maximum  9  How 
can  we  know  that  this  maximum  is  reached  and  that  the 
steam  is  used  in  the  most  advantageous  way  possible  to  produce 
motive  power  ? 

Since  any  re-establishment  of  equilibrium  in  the  caloric  can 
be  used  to  produce  motive  power,  any  re-establishment  of  equi- 
librium which  is  effected  without  producing  motive  power 
should  be  considered  as  a  veritable  loss:  now,  with  little  re- 
flection, we  can  see  that  any  change  of  temperature  which  is  not 
due  to  a  change  of  volume  of  the  body  can  be  only  a  useless  re- 
establishment  of  equilibrium  in  the  caloric.*  The  necessary 
condition  of  the  maximum  is,  then,  that  in  bodies  used  to  obtain 
the  motive  power  of  heat,  no  change  of  temperature  occurs  which  is 

has  ever  been  produced,  meaning  by  that,  a  motion  which  continues  in- 
definitely without  change  in  the  body  used  as  an  agent  ? 

The  electromotive  apparatus  (Volta's  pile)  has  sometimes  been  considered 
capable  of  producing  perpetual  motion;  the  attempt  has  been  made  to  re- 
alize it  by  the  construction  of  the  dry  pile,  which  is  claimed  to  be  unal- 
terable ;  but,  in  spite  of  all  that  has  been  done,  the  apparatus  always  dete- 
riorates perceptibly  when  its  action  is  sustained  for  some  time  with  any 
energy. 

The  general  and  philosophical  acceptation  of  the  words  perpetual  motion 
should  comprehend  not  only  a  motion  capable  of  indefinite  continuance 
after  it  has  been  started,  but  also  the  action  of  an  apparatus,  of  a  set  of 
bodies,  capable  of  creating  motive  power  in  an  unlimited  quantity,  and  of 
setting  in  motion  successively  all  the  bodies  of  nature,  if  they  are  originally 
at  rest,  and  of  destroying  in  them  the  principle  of  inertia,  and  finally  capa- 
ble of  furnishing  in  itself  all  the  forces  necessary  to  move  the  entire  uni- 
verse, to  prolong  and  to  constantly  accelerate  its  motion.  Such  would  be 
a  real  creation  of  motive  power.  If  this  were  possible,  it  would  be  useless 
to  search  for  motive  power  in  combustibles,  in  currents  of  water  and 
air.  We  should  have  at  our  disposal  an  inexhaustible  source  from  which 
we  could  draw  at  will. 

*  We  do  not  here  take  into  consideration  any  chemical  action  between  the 
bodies  used  to  obtain  the  motive  power  of  heat.  The  chemical  action  which 
occurs  in  the  source  is  in  a  sense  preliminary,  an  fiction  not  designed  to  im- 
mediately create  motive  power,  but  to  destroy  equilibrium  in  the  caloric,  to 
produce  a  difference  in  temperature  which  shall  finally  result  in  motion. 
13 


MEMOIRS    ON 

not  due  to  a  change  of  volume.  Conversely,  every  time  that  this 
oon.lition  is  fulfilled,  the  maximum  is  attained. 

This  principle  should  not  be  lost  sight  of  in  the  construction 
of  heat-engines.  It  is  the  foundation  upon  which  they  rest.  If 
it  cannot  be  rigorously  observed,  it  should  at  least  be  departed 
from  as  little  as  possible. 

Any  change  of  temperature  which  is  not  due  to  a  change  of 
volume  or  to  chemical  action  (which  we  provisionally  assume  not 
to  occur  in  this  case)  is  necessarily  due  to  the  direct  transfer  of 
caloric  from  a  hotter  to  a  colder  body.  This  transfer  takes 
place  principally  at  the  points  of  contact  of  bodies  at  different 
temperatures;  thus  such  contacts  should  be  avoided  as  much 
as  possible.  They  doubtless  cannot  be  avoided  entirely,  but  at 
least  care  should  be  taken  that  the  bodies  brought  in  contact 
should  differ  but  little  in  temperature. 

When  we  assumed  in  the  previous  demonstration  that  the  ca- 
loric of  the  body  A  was  used  to  produce  steam,  we  supposed  the 
steam  to  be  produced  at  the  same  temperature  as  that  of  the 
body  A;  thus  the  only  contact  was  between  two  bodies  of  equal 
temperature ;  the  change  of  temperature  which  the  steam  after- 
wards experienced  was  due  to  expansion  and  consequently  to  a 
change  of  volume;  finally  condensation  was  effected  without 
contact  of  bodies  of  different  temperatures.  It  was  effected  by 
the  exercise  of  a  constant  pressure  on  the  steam  brought  in  con- 
tact with  the  body  B,  at  the  same  temperature  as  that  of  the 
body  It.  The  condition  of  the  maximum  was  thus  fulfilled.  In 
reality  things  would  not  occur  exactly  as  we  have  supposed.  I  n 
order  to  effect  a  transfer  of  the  caloric  from  one  body  to  the 
other,  the  first  must  have  the  higher  temperature  ;  but  this  dif- 
ference may  be  supposed  to  be  as  small  an  we  please  ;  wo  may, 
in  theory,  consider  it  zero  without  invalidating  the  arirumt -nt. 

A  more  valid  objection  may  be  made  to  our  demonstration, 
namely: 

When  we  produce  steam  by  taking  caloric  from  the  body  .1. 
Mini  when  this  steam  is  afterward  condensed  by  contact  with 
the  body  h,  the  water  used  to  form  it,  which  was  a>snmed  to  In-. 
at  the  beginning,  at  the  temperature  of  the  \»»\\  .  I.  i-.  at  the 
end  of  the  operation,  at  the  temperature  of  the  hody  IS — that  is. 
it  is  colder.  If  we  wish  to  recommence  an  operation  similar  to 
the  first,  to  develop  a  new  quantity  of  motive  power  with  the 
same  instrument  and  the  same  steam,  we  must  first  re-establish 
14 


THE    SECOND    LAW    OF    THERMODYNAMICS 

the  original  state  of  things  and  bring  the  water  to  the  temper- 
ature which  it  had  at  first.  This  can  no  doubt  be  done  by 
placing  it  immediately  in  contact  with  the  body  A  ;  but  in  that 
case  there  is  contact  between  bodies  of  different  temperatures 
and  loss  of  motive  power.*  It  would  become  impossible  to 
perform  the  reverse  operation — that  is,  to  cause  the  caloric 
used  in  raising  the  temperature  of  the  liquid  to  return  to  the 
body  A. 

This  difficulty  can  be  removed  by  supposing  the  difference  of 
temperature  between  the  body  A  and  the  body  B  infinitely 
small ;  the  quantity  of  heat  needed  to  bring  the  liquid  back  to 
its  original  temperature  is  also  infinitely  small  and  negligible 
relatively  to  that  finite  quantity  which  is  needed  to  produce  the 
steam. 

The  proposition  being  thus  demonstrated  for  the  case  in  which 
the  difference  of  temperature  of  the  two  bodies  is  infinitely 
small  may  easily  be  extended  to  cover  the  general  case.  In  fact, 
if  we  desire  to  produce  motive  power  by  the  transfer  of  caloric 
from  the  body  A  to  the  body  Z,  the  temperature  of  the  latter 
body  being  very  different  from  that  of  the  former,  we  may 
imagine  a  series  of  bodies  B,  C,  D  .  .  .at  temperatures  interme- 
diate between  those  of  the  bodies  A  and  Z,  and  chosen  in  such 
a  manner  that  the  differences  between  A  and  B,  B  and  C  .  .  . 
shall  be  always  infinitely  small.  The  caloric  which  proceeds 
from  A  arrives  at  Z  only  after  having  passed  through  the  bodies 
B,  C,  D  .  .  .  and  after  having  developed  in  each  of  these  trans- 
fers the  maximum  of  motive  power.  The  reverse  operations  are 
here  all  possible,  and  the  reasoning  on  page  11  becomes  rigor- 
ously applicable. 

According  to  the  views  now  established  we  may  with  pro- 

*This  kind  of  loss  is  always  met  with  in  steam-engines.  In  fact,,  the 
water  which  supplies  the  boiler  is  always  colder  than  that  which  it  already 
contains,  and  hence  a  useless  re-establishment  of  equilibrium  in  the  ca- 
loric takes  place  between  them.  It  is  easy  to  see  a  posteriori  that  this  re- 
estjiblishment  of  equilibrium  entails  a  loss  of  motive  power  if  we  reflect 
that  it  would  be  possible  to  heat  the  water  supply  before  injecting  it  by 
using  it  as  water  of  condensation  in  a  small  accessory  engine,  in  which 
stenm  taken  from  the  large  boiler  could  be  used  and  in  which  condensation 
would  occur  at  a  temperature  intermediate  between  that  of  the  boiler  and 
that  of  the  principal  condenser.  The  force  produced  by  the  small  engine 
would  entail  no  expenditure  of  heat,  since  all  that  it  would  use  would  re- 
cuter  the  boiler  with  the  water  of  condensation. 
15 


MEMOIRS    ON 

priety  compare  the  motive  power  of  heat  with  that  of  a  water- 
fall; both  have  a  maximum  which  cannot  be  surpassed,  whatever 
may  be,  on  the  one  hand,  the  machine  used  to  receive  the  action 
of  the  water  and  whatever,  on  the  other  hand,  the  substance 
used  to  receive  the  action  of  the  heat.  The  motive  power  of 
fulling  water  depends  on  the  quantity  of  water  and  on  the 
height  of  its  fall;  the  motive  power  of  heat  depends  also  on  the 
quantity  of  caloric  employed  and  on  that  which  might  be  named, 
which  we,  in  fact,  will  call,  its  descent* — that  is  to  say.  on  t  he  dif- 
ference of  temperature  of  the  bodies  between  which  t  he  exchange 
of  caloric  is  effected.  In  the  fall  of  water  the  motive  power  is 
strictly  proportional  to  the  difference  of  level  between  the  higher 
and  lower  reservoirs.  In  the  fall  of  caloric  the  motive  power 
doubtless  increases  with  the  difference  of  temperature  between 
the  hotter  and  colder  bodies,  but  we  do  not  know  whether  it  is 
proportional  to  this  difference.  We  do  not  know,  for  example, 
whether  the  fall  of  the  caloric  from  100  to  50  degrees  furnishes 
more  or  less  motive  power  than  the  fall  of  the  same  caloric  from 
50  degrees  to  zero.  This  is  a  question  which  we  propose  to  ex- 
amine later. 

We  shall  give  here  a  second  demonstration  of  the  funda- 
mental proposition  stated  on  page  13  and  present  this  propo- 
sition in  a  more  general  form  than  we  have  before. 

When  a  gaseous  fluid  is  rapidly  compressed  its  temperature 
rises,  and  when  it  is  rapidly  expanded  its  temperature  falls. 
This  is  one  of  the  best  established  facts  of  experience ;  we  shall 
take  it  as  the  basis  of  our  demonstration,  f  When  tin  tem- 
perature of  a  gas  is  raised  and  we  wish  to  bring  it  back  to  its 

•The  matter  here  treated  being  entirely  new.  we  are  obliged  to  employ 
expressions  hitherto  unused  and  which  are  not  perhaps  as  clear  as  could  be 
desired. 

f  The  facts  of  experience  which  best  prove  the  change  of  temperature 
of  a  gas  by  compression  or  expansion  are  the  following  : 

1.  The  fall  of  temperature  indicated  by  a  thermometer  placed  under  tin- 
receiver  of  an  air  pump  in  which  a  \.n  mini  is  produced.     This  is  very  p<  t 
rrptililr  with  a  Bregucl  thermometer  ;  it  may  amount  to  upwards  of  40  or 
50  degrees.    The  cloud,  which  is  formed  in  this  operation  seems  to  in  •tin  to 
ii,c  c»ii<lriisalion  of  water  vapor  caii-cil  hy  the  coolintr  of  the  air. 

2.  The  ignition  of  tinder  in  the  so-called  fire-syringe  (pneumatic  tinder- 
box),  which  is,  as  is  well  known,  a  small  pump  in  which  air  may  !><•  rapidly 
compressed. 

8.  The  fall  of  temperature  indicated  by  a  thermometer  placed  in  a  re- 
16 


THE   SECOND    LAW    OF   THERMODYNAMICS 

original  temperature  without  again  changing  its  volume,  it  is 
necessary  to  remove  caloric  from  it.  This  caloric  may  also  be 
removed  as  the  compression  is  effected,  so  that  the  temperature 
of  the  gas  remains  constant.  In  the  same  way,  if  the  gas  is 
rarefied,  we  can  prevent  its  temperature  from  falling,  by  fur- 
nishing it  with  a  certain  quantity  of  caloric.  "We  shall  call  the 
caloric  used  in  such  cases,  when  it  occasions  no  change  of  tem- 
perature, caloric  due  to  a  change  of  volume.  This  name  does  not 
indicate  that  the  caloric  belongs  to  the  volume;  it  does  not  be- 
long to  it  any  more  than  it  does  to  the  pressure,  and  it  might 
equally  well  be  called  caloric  due  to  a  change  of  pressure.  We  are 
ignorant  of  what  laws  it  obeys  in  respect  to  changes  of  volume : 
it  is  possible  that  its  quantity  changes  with  the  nature  of  the 

ceptacle  in  which  air  has  been  compressed,  and  from  which  it  is  allowed  to 
escape  by  opening  a  stopcock. 

4.  The  results  of  experiments  on  the  velocity  of  sound.  M.  de  Laplace 
has  shown  that  to  harmonize  these  results  with  theory  and  calculation  we 
must  assume  that  air  is  heated  by  a  sudden  compression. 

The  only  fact  which  can  be  opposed  to  these  is  an  experiment  of  MM. 
Gay-Lussac  and  Welter,  described  in  the  Annales  de  Chimie  et  de  Physique. 
If  a  small  opening  is  made  in  a  large  reservoir  of  compressed  air,  and  the 
bulb  of  a  thermometer  is  placed  in  the  current  of  air  escaping  through  this 
opening,  no  perceptible  fall  of  temperature  is  indicated  by  the  thermometer. 

We  may  explain  this  fact  in  two  ways  : 

1.  The  friction  of  the  air  against  the  walls  of  the  opening  through  which 
it  escapes  may  perhaps  develop  enough  heat  to  be  noticed  ;  2.  The  air 
which  impinges  immediately  upon  the  bulb  of  the  thermometer  may  re- 
cover by  its  shock  against  the  bulb,  or  rather  by  the  detour  which  it  is 
forced  to  make  by  the  encounter,  a  density  equal  to  that  which  it  had  in 
the  receiver,  somewhat  in  the  same  way  as  a  current  of  water  rises  above 
its  level  when  it  meets  a  fixed  obstacle. 

The  change  of  temperature  in  gases  occasioned  by  a  change  of  volume 
may  be  considered  one  of  the  most  important  facts  in  physics,  because  of 
the  innumerable  consequences  which  it  entails,  and  at  the  same  time  as  one 
of  the  most  difficult  to  elucidate  and  to  measure  by  conclusive  experi- 
ments. It  presents  singular  anomalies  in  several  cases. 

Must  we  not  attribute  the  coldness  of  the  air  in  high  regions  of  the  at- 
mosphere to  the  lowering  of  its  temperature  by  expansion  ?  The  reasons 
hitherto  given  to  explain  this  coldness  are  entirely  insufficient ;  it  has  been 
said  that  the  air  in  high  regions,  receiving  but  a  small  amount  of  heat 
reflected  by  the  earth,  and  itself  radiating  into  celestial  space,  would  lose 
caloric  and  thus  become  colder  ;  but  this  explanation  is  overthrown  when 
we  consider  that  at  equal  elevations  the  cold  is  as  great  or  even  greater  on 
elevated  plains  than  on  the  tops  of  mountains  or  in  parts  of  the  atmosphere 
distant  from  the  earth. 

B  17 


MEMOIRS    ON 


gas,  or  with  its  density  or  with  its  temperature.  Experiment 
has  tanght  us  nothing  on  this  subject;  it  has  taught  us  only 
that  this  caloric  is  developed  in  greater  or  less  quantity  by  the 
compression  of  elastic  fluids. 

This  preliminary  idea  having  been  stated,  let  us  imagine  an 
elastic  fluid — atmospheric  air,  for  example — enclosed  in  a  cylin- 
drical vessel  abed  (Fig.  1)  furnished  with  a  movable  diaphragm 
or  piston  cd;   let  us  assume  also  the 
two  bodies  A,  B  both  at  constant  tem- 
peratures, that  of  A  being  higher  than 
that  of  B,  and  let  us  consider  the  series 
of  operations  which  follow  : 

1.  Contact  of  the  body  A  with  the 
air  contained  in  the  vessel  ///«•»/  <>r  with 
the  wall  of  this  vessel,  which  wall  is 
supposed  to  be  a  good  conductor  of 
caloric.     By  means  of  this  contact  the 
air  attains  the  same  temperature  as  the 
body  .4;  cd  is  the  position  of  the  pis- 
ton. 

2.  The  piston  rises  gradually  until 
it  takes  the  position  ef.     Contact  is  al- 
ways maintained  between  the  air  ami 

flKrr^          «™«^      the  body  A,  and  the  temperature  thus 
1  remains   constant  during  the  rarefac- 

tion.    The  body  A  furnishes  the  ca- 
loric necessary  to  maintain  a  constant 
Fig.  i  temperature. 

3.  The  body  A  is  removed  and  the 

air  is  no  longer  in  contact  with  any  body  capable  of  supply- 
ing it  with  caloric;  the  piston,  however,  continues  to  move 
and  passes  from  the  position  ef  to  the  position  gh.     The  air  is 
rarefied  without  receiving  caloric  and   its  temperature  falls. 
Let  us  suppose  that  it  falls  until  it  becomes  equal  to  that  of 
the  body  />';  at  this  instant  the  piston  ceases  to  move  and 
occupies  the  position  ////. 

4.  The  air  is  brought  in  contact  with  the  body  /.' :  it  is  com- 
pressed by  the  piston  as  it  returns  from  the  position  ////  to  the 
position  nl.  The  air,  however,  remains  at  a  constant  tempera- 
ture on  account  of  its  contact  with  the  body  B,  to  which  it  gives 
up  its  caloric. 

18 


THE  SECOND  LAW  OF  THERMODYNAMICS 

5.  The  body  B  is  removed  and  the  compression  of  the  air  con- 
tinued.    The  temperature  of  the  air,  which  is  now  isolated, 
rises.     The  compression  is  continued  until  the  air  acquires  the 
temperature  of  the  body  A.     The  piston  during  this  time  passes 
from  the  position  cd  to  the  position  ik. 

6.  The  air  is  again  brought  in  contact  with  the  body  A\  the 
piston  returns  from  the  position  ik  to  the  position  ef,  and  the 
temperature  remains  constant. 

7.  The  operation  described  in  No.  3  is  repeated,  and  then  the 
operations  4,  5,  6,  3,  4,  5,  6,  3,  4,  5,  and  so  on,  successively. 

In  these  various  operations  a  pressure  is  exerted  upon  the 
piston  by  the  air  contained  in  the  cylinder ;  the  elastic  force 
of  this  air  varies  with  the  changes  of  volume  as  well  as  with 
the  changes  of  temperature  ;  but  we  should  notice  that  at  equal 
volumes — that  is,  for  similar  positions  of  the  piston — the  tem- 
perature is  higher  during  the  expansions  than  during  the  com- 
pressions. During  the  former,  therefore,  the  elastic  force  of 
the  air  is  greater,  and  consequently  the  quantity  of  motive 
power  produced  by  the  expansions  is  greater  than  that  which  is 
consumed  in  effecting  the  compressions.  Thus  there  remains 
an  excess  of  motive  power,  which  we  can  dispose  of  for  any 
purpose  whatsoever.  The  air  has  therefore  served  as  a  heat-en- 
gine ;  and  it  has  been  used  in  the  most  advantageous  way  pos- 
sible, for  there  has  been  no  useless  re-establishment  of  equilib- 
rium in  the  caloric. 

All  the  operations  described  above  can  be  carried  out  in  a 
direct  and  in  a  reverse  order.  Let  us  suppose  that  after  the 
sixth  step,  when  the  piston  is  at  ef,  it  is  brought  back  to  the 
position  ik,  and  that,  at  the  same  time,  the  air  is  kept  in  con- 
tact with  the  body  A  ;  the  caloric  furnished  by  this  body  dur- 
ing the  sixth  operation  returns  to  its  source — that  is,  to  the  body 
A — and  the  condition  of  things  is  the  same  as  at  the  end  of  the 
fifth  operation.  If  now  we  remove  the  body  A  and  move  the 
piston  from  ef  to  cd,  the  temperature  of  the  air  will  fall  as 
many  degrees  as  it  rose  during  the  fifth  operation  and  will  equal 
that  of  the  body  B.  A  series  of  reverse  operations  to  those 
above  described  could  evidently  be  carried  out ;  it  is  only  neces- 
sary to  bring  the  system  into  the  same  initial  state  and  in  each 
operation  to  carry  out  an  expansion  instead  of  a  compression, 
and  conversely. 

The  result  of  the  first  operation  was  the  production  of  a  cer- 
19 


MKMOIRS    ON 

tain  quantity  of  motive  power  and  the  transfer  of  the  caloric 
from  the  body  -1  to  the  body  B  ;  the  result  of  the  reverse  opera- 
tion  would  be  the  consumption  of  the  motive  power  product-  1 
and  the  return  of  the  caloric  from  the  body  B  to  the  body  .1  : 
so  that  the  two  series  of  operations  in  a  sense  annul  or  neutralize 
each  other. 

The  impossibility  of  making  the  caloric  produce  a  larger 
quantity  of  motive  power  than  that  which  we  obtained  in  our 
first  series  of  operations  is  now  easy  to  prove.  It  may  be  de- 
monstrated by  an  argument  similar  to  that  used  on  pa  ire  11. 
The  argument  will  have  even  a  greater  degree  of  rigor  :  the  air 
which  serves  to  develop  the  motive  power  is  brought  back, 
at  the  end  of  each  cycle  of  operations,  to  its  original  condi- 
tion, which  was,  as  we  noticed,  not  quite  the  case  with  the 
steam.* 

We  have  chosen  atmospheric  air  as  the  agency  employed  t.. 
develop  the  motive  power  of  heat;  but  it  is  evident  that  the 
same  reasoning  would  hold  for  any  other  gaseous  substance,  and 
even  for  all  other  bodies  susceptible  of  changes  of  temperature 
by  successive  contractions  and  expansions  —  that  is,  for  all 
bodies  in  Nature,  at  least,  all  those  which  are  capable  of  develop- 
ing the  motive  power  of  heat.  Thus  we  are  led  to  establish  this 
general  proposition : 

The  motive  power  \  of  /ie<tt  i*  imli'i>i->nlrnt  of  the  ay> 

•We  implicitly  assume-,  in  our  demonstration.  tli.-U  if  a  body  experi- 
cnces  any  changes,  ami  returns  exactly  to  its  origin  tl  stair.  vt.  t  a  certain 
numtterof  transformations — thnl  is  to  say,  to  its  original  Mate  determined  by 
its  density,  its  temperature,  and  its  mode  of  aegrcpitioii  :  \v. •  a-Mime.  I  say, 
that  the  body  contains  the  same  quantity  of  hc:it  as  it  contained  nl  first,  or. 
in  other  words,  thatthcqtinniitiesof  heat  absorbed  and  te|.-as<  d  in  ii- 
transformations  exactly  compensate  one  another.  Thi  fact  has  never  IN-CM 
called  in  question  ;  it  was ftt  first  admitted  wiihoul  consider  iti-m  and  after 
wanis  verified  in  many  cases  by  exporimcnts  with  the  ralorim- -IIT.  To 
deny  it  would  he  to  overthrow  UK*  entire  theory  of  heat,  of  wliich  it  is  die 
i  foundation.  It  may  IK;  rcmarkc.l,  in  passinp.  that  tin-  fiin<l:inicntal  prin 
i-ipli-soii  which  the  theory  of  heat  rests  should  l«  jfiven  the  nvM  cueful 
i  \  itnination.  Severnl  experimental  fat  is  si-<-m  to  Ix;  almost  inexplicable  in 
Hi-  actual  Htaic  of  that  theory.  [The  doubts  here  expressed  as  to  the 
validity  of  the  assumptions  made  with  respect  to  the  nature  of  li 
vclo|M*d  in  (-arnot's  mind  into  an  actual  rejection  of  those  assumption*,  an. I 
led  him  to  suspect  the  true  nature  of  heat.  See  Life  </  ' '  '  KM.  ] 

f  [//  may  be  well  to  notife  a<j,iiit   t/i.it  Curnnt  utet  "  motir, 
lyminymout  ifilh  the  more  modern  term  "  work."\ 

m 


THE   SECOND   LAW   OF   THERMODYNAMICS 

ployed  to  develop  it ;  its  quantity  is  determined  solely  by  the  tem- 
peratures of  the  bodies  between  which,  in  the  final  result,*  the 
transfer  of  the  caloric  occurs. 

It  is  understood  in  this  statement  that  the  method  used  for 
developing  motive  power,  whatever  it  may  be,  attains  the  highest 
perfection  of  which  it  is  capable.  This  condition  will  be  ful- 
filled, as  we  remarked  above,  if  there  is  no  change  of  tempera- 
ture in  the  bodies  which  is  not  due  to  a  change  of  volume  or, 
which  amounts  to  the  same  thing  differently  expressed,  if  the 
temperatures  of  the  bodies  which  come  in  contact  with  each 
other  are  never  perceptibly  different. 

Various  methods  of  developing  motive  power  may  be  adopted, 
either  by  the  use  of  different  substances  or  of  the  same  sub- 
stance in  different  states;  for  example,  by  the  use  of  a  gas  at 
two  different  densities. 

This  remark  leads  us  naturally  to  the  interesting  study  of  aeri- 
form fluids,  a  study  which  will  conduct  us  to  new  results  con- 
cerning the  motive  power  of  heat,  and  will  give  us  the  means 
of  verifying  in  some  particular  cases  the  fundamental  proposi- 
tion stated  above,  f 

It  can  easily  be  seen  that  our  demonstration  will  be  simplified 
if  we  suppose  the  temperatures  of  the  bodies  A  and  B  to  be  very 
slightly  different.  Then  the  movements  of  the  piston  will 
be  very  small  during  operations  3  and  5,  and  these  operations 
may  be  suppressed  without  perceptible  influence  on  the  de- 
velopment of  motive  power.  That  is,  a  very  small  change  of 
volume  ought  to  be  sufficient  to  produce  a  very  small  change 
of  temperature,  and  this  change  of  volume  is  negligible  com- 
pared with  that  of  operations  4  and  6,  which  are  unrestricted 
in  extent. 

If  we  suppress  operations  3  and  5  in  the  series  above  de- 
scribed, it  is  reduced  to  the  following: 

1.  Contact  of  the  gas  contained  in  abed  (Fig.  2)  with  the  body 
A,  and  passage  of  the  piston  from  cd  to  ef; 

2.  Removal  of  the  body  A,  contact  of  the  gas  enclosed  in  abcf 
with  the  body  B,  and  return  of  the  piston  from  ef  to  cd  ; 

3.  Removal  of  the  body  B,  contact  of  the  gas  with  the  body 


*  [That  is,  upon  tlie  completion  of  a  cycle  of  operations.] 
\  We  shall  suppose  in  what  follows  that  the  reader  is  familiar  with  the 
latest  progress  of  modern  physics  in  the  departments  of  heat  and  gases. 
21 


1  KMOIRS    ON 


A,  and  passage  of  the  piston  from  cd  to  ef—  that  is  to  say, 
tition  of  the  first  operation,  and  so  on. 

The  motive  power  resulting  from  the  operations  1,  2,  3,  taken 
together,  will  evidently  be  the  difference  between  that  which 
is  produced  by  the  expansion  of  the  gas  while  its  temperature 
equals  that  of  the  body  A  and  that  which  is  consumed  to  com- 
press the  gas  while  its  temperature  equals  that  of  the  body  B. 

Let  us  suppose  that  the  operations  1  and  'I  are  performed 
with  two  gases  which  are  chemically  different,  but  which  are 
subjected  to  the  same  pressure  —  for  example,  that  of  the  atiuos- 


Pig. 


phere  ;  these  gases  behave  in  the  same  circumstances  in  exactly 
the  same  way — that  is  to  say,  their  expansive  forces,  originally 
equal,  remain  so  irrespective  of  changes  of  volume  ami  temper- 
ature, provided  that  these  changes  are  the  same  in  both.  This 
is  an  evident  consequence  of  the  laws  of  Mariotte  ami  <>f  M  M. 
<ia  v-Lussiic  and  Dalton,  which  laws  are  eommoii  t<>  all  elastic 
fluids,  and  in  virtue  of  which  the  same  relation-  exist  in  all 
these  fluids  between  the  volume,  expansive  force,  ami  temper- 
ature. Since  two  different  gases,  taken  at  the  same  tempera- 
ture and  under  the  same  pressure,  should  liehave  alike  under 
the  same  circumstances,  they  should  produce  eijual  quantities 
of  motive  power  when  subjected  to  the  operations  above  de- 
scribed. Now  this  implies,  according  to  the  fundamental 
proposition  which  we  have  established,  that  two  equal  quanti- 
ties of  caloric  are  employed  in  these  operations— that  is,  that 
the  quantity  of  caloric  transferred  from  the  body  A  to  the  body 


THE   SECOND   LAW    OF   THERMODYNAMICS 

B  is  the  same  whichever  of  the  two  gases  is  used  in  the  opera- 
tions. The  quantity  of  caloric  transferred  from  the  body  A  to 
the  body  B  is  evidently  that  which  is  absorbed  by  the  gas  in 
the  increase  of  its  volume,  or  that  which  it  afterwards  emits 
during  compression.  We  are  thus  led  to  lay  down  the  follow- 
ing proposition  : 

When  a  gas  passes  without  change  of  temperature  from  one  defi- 
nite volume  and  pressure  to  another,  the  quantity  of  caloric  ab- 
sorbed or  emitted  is  alivays  the  same,  irrespective  of  the  nature  of 
the  gas  chosen  as  the  subject  of  the  experiment. 

For  example,  consider  1  litre  of  air  at  the  temperature  of 
100  degrees  and  under  the  pressure  of  1  atmosphere*,  if  the 
volume  of  this  air  is  doubled,  a  certain  quantity  of  heat  must 
be  supplied  to  it  in  order  to  maintain  it  at  the  temperature  of 
100  degrees.  This  quantity  will  be  exactly  the  same  if,  instead 
of  performing  the  operation  with  air,  we  use  carbonic  acid  gas, 
nitrogen,  hydrogen,  vapor  of  water,  or  of  alcohol — that  is,  if  we 
double  the  volume  of  1  litre  of  any  one  of  these  gases  at  the 
temperature  of  100  degrees  and  under  atmospheric  pressure. 

The  same  thing  would  be  true,  in  the  reverse  sense,  if  the 
volume  of  the  gas,  instead  of  being  doubled,  were  reduced  one- 
half  by  compression. 

The  quantity  of  heat  absorbed  or  set  free  by  elastic  fluids 
during  their  changes  of  volume  has  never  been  measured  by 
direct  experiment.  Such  an  experiment  would  doubtless  pre- 
sent great  difficulties,  but  we  have  one  result  which  for  our  pur- 
poses is  nearly  equivalent  to  it ;  this  result  has  been  furnished 
by  the  theory  of  sound,  and  may  be  received  with  confidence  be- 
cause of  the  rigor  of  the  demonstration  by  which  it  has  been 
established.  It  may  be  described  as  follows  : 

Atmospheric  air  will  rise  in  temperature  1  degree  centigrade 
when  its  volume  is  reduced  by  y|^  by  sudden  compression.* 

The  experiments  on  the  velocity  of  sound  were  made  in  air 
under  a  pressure  of  760  millimetres  of  mercury  and  at  the  tem- 
perature of  G  degrees ;  and  it  is  only  in  these  circumstances  that 
Poisson's  statement  is  applicable.  We  shall,  however,  for  the 

*  M.  Poisson,  to  whom  we  owe  this  statement,  has  shown  that  it  agrees 
very  well  with  the  results  of  an  experiment  by  MM.  Clement  and  Desormes 
on  the  behavior  of  air  entering  into  a  vacuum  or  rather  into  slightly  rare- 
fied air.  It  agrees  also,  very  nearly,  with  a  result  obtained  by  MM.  Gay- 
Lussac  and  Welter.  (See  note,  p.  32.) 


MEMOIRS    ON 

sake  of  convenience,  consider  it  to  hold  at  a  tcmperatnre  of  0 
degrees,  which  is  only  slightly  different. 

Air  compressed  by  -pj-y  and  so  raised  in  temperature  1  degree 
differs  from  air  heated  directly  by  the  same  amount  only  in  its 
density.  If  we  call  the  original  volume  V,  the  compression  1>\ 
Y^-y  reduces  it  to  V—  'j\j  V.  Direct  heating  under  constant 
pressure,  according  to  the  law  of  M.  Gay-Lussuc,  should  in- 
crease the  volume  of  the  air  by  ^  of  that  which  it  would  have 
at  0  degrees;  thus  the  volume  of  the  air  is  in  one  process  re- 
duced to  V—  -rfj  T",  and  in  the  other  increased  to  V+  5^-7  T. 
The  difference  between  the  quantities  of  heat  present  in  tin- 
air  in  the  two  cases  is  evidently  the  quantity  used  to  raise  its 
temperature  directly  by  1  degree  ;  thus  the  quantity  of  heat  ab- 
sorbed by  the  air  in  passing  from  the  volume  I'—  ,  ],.,  V  to  the 
volume  r+^Tis  equal  to  that  which  is  necessary  to  raise 
its  temperature  1  degree. 

Let  us  now  suppose  that,  instead  of  heating  the  air  while  sub- 
jected to  a  constant  pressure  and  able  to  expand  freely,  we  en- 
close it  in  an  envelope  not  capable  of  expansion,  and  then  raise 
its  temperature  1  degree.  The  air  thus  heated  1  degree  dif- 
fers from  air  compressed  by  yfy,  by  having  its  volume  larger 
by  y^.  Thus,  then,  the  quantity  of  heat  which  the  air  gives  up 
by  a  reduction  of  its  volume  by  jfa  is  equal  to  that  which  is  re- 
quired to  raise  its  temperature  1  degree  at  constant  volume. 
As  the  differences,  V—  TfyJr,  F,  and  F+?$T  I  .  are  .-mall  in 
comparison  with  the  volumes  themselves,  we  may  consider  the 
quantities  of  heat  absorbed  by  the  air  in  passing  from  the  first 
of  these  volumes  to  the  second,  and  from  the  first  to  the  third. 
M  sensibly  proportional  to  the  changes  of  volume.  \\  e  thus 
obtain  the  following  relation  : 

The  quantity  of  heat  required  to  raise  the  temperature  of  air 
under  constant  pressure  1  degree  is  to  the  quantity  required  to 
raise  it  1  degree  at  constant  volume  in  the  ratio  of  the  numbers 

rir  +  jfr  to  rfy, 
or.  multiplying  both  terms  by  11G.2G7,  in  the  ratio  of  the  nnin- 

;7+116  to  267. 

This  is  the  ratio  between  the  capacity  for  heat  of  air  under 
constant  pressure  and  its  capacity  at  constant  volume.  If  tin- 
first  of  these  two  capacities  is  expressed  by  unity  the  other  will 

be  expressed  by  the  number  -  -      "        <» •.  approximately,  0.700. 

•Ji 


THE   SECOND   LAW   OF   THERMODYNAMICS 

Their  difference — 1—0.700  or  0.300 — will  evidently  express  the 
quantity  of  heat  which  will  occasion  the  increase  of  volume  of 
the  air  when  its  temperature  is  raised  1  degree  under  constant 
pressure. 

From  the  law  of  MM.  Gay-Lussac  and  Daltou  this  increase 
of  volume  will  be  the  same  for  all  other  gases ;  from  the  the- 
orem demonstrated  on  page  23  the  heat  absorbed  by  equal  in- 
crements of  volume  is  the  same  for  all  elastic  fluids;  we  are 
thus  led  to  establish  the  following  proposition  : 

The  difference  between  the  specific  heat  under  constant  pressure 
and  the  specific  heat  at  constant  volume  is  the  same  for  all  gases. 

It  must  be  noticed  here  that  all  the  gases  are  assumed  to  be 
taken  at  the  same  pressure — for  example,  the  pressure  of  the 
atmosphere — and  also  that  the  specific  heats  are  measured  in 
terms  of  the  volumes. 

Nothing  is  now  easier  than  to  construct  a  table  of  the  specific 
heats  of  gases  at  constant  volume  with  the  aid  of  our  knowl- 
edge of  their  specific  heats  under  constant  pressure.  We 
present  this  table,  the  first  column  of  which  contains  the  re- 
sults of  direct  experiments  by  MM.  Delaroche  and  Berard  on 
the  specific  heat  of  gases  under  atmospheric  pressure.  The 
second  column  contains  the  numbers  in  the  first  diminished  by 
0.300. 

TABLE  OF  THE  SPECIFIC  HEAT  OF  GASES 


GASES 

SPECIFIC  HEAT  UN- 
DER CONSTANT 
PRESSURE 

SPECIFIC  HEAT 
AT  CONSTANT 
VOLUME 

Atmospheric  air 

1  000 

0  700 

Hydrogen 

0  903 

0  603 

Carbonic  acid  . 

1  258 

0  958 

Oxygen  .  . 

0.976 

0  676 

Nitrogen  .  .      .  .        

1  000 

0  700 

Nitrons  oxide 

1  350 

1  050 

Olefiant  gas  

1.553 

1  253 

Carbonic  oxide  .  . 

1.034 

0.734 

The  numbers  in  the  two  columns  are  referred  to  the  same 
unit,  to  the  specific  heat  of  atmospheric  air  under  constant 
pressure. 

The  difference  between  the  corresponding  numbers  in  the 
two  columns  being  constant,  the  ratio  between  them  should  be 
25 


MEMOIRS    ON 

variable  ;  thus  the  ratio  between  the  specific  heats  of  gases  un- 
der constant  pressure  and  at  constant  volume  varies  for  the 
different  gases. 

\\  f  have  seen  that  the  temperature  of  the  air  when  it  under- 
goes a  sudden  compression  of  t  fff  of  its  volume  rises  1  degree. 
That  of  other  gases  should  also  rise  when  they  are  similarly 
roiii|>ivss<_-<l.  The  temperature  should  rise,  not  equally  for  all, 
but  in  the  inverse  ratio  of  their  specific  heats  at  constant 
volume.  In  fact,  the  reduction  of  volume  being,  by  hypothesis, 
always  the  .-aim-,  the  quantity  of  heat  due  to  this  reduction 
should  also  be  always  the  same,  and  consequently  should  cans.' 
a  rise  of  temperature  depending  only  on  the  specific  heat  of  the 
gas  after  its  compression,  and  evidently  in  an  inverse  ratio  to 
that  specific  heat.  It  is  therefore  easy  to  construct  the  table 
of  elevations  of  temperature  of  the  different  gases  for  a  com- 
pression 


TABLE   OF  THE   ELEVATION    OF   Till:   TKM  I'KKATl  UK   OF   &A8E8 
1H  K  TO  COMPRESSION 

ELEVATION    OK  TKMi'i  I:  \ 

GASES  TI-KK  Kon  A  in  in  .  noa 

OF  VOLfMK  OK  ,\f 

o 

Atmospheric  air  ........................  1.000 

Hydrogen  ............................  1.160 

Carbonic  acid  ..........................  0.730 

Oxygen  ................................  1.035 

Nitrogen  ..............................  1.000 

Nitrous  oxide  ..........................  0.< 

Olefiant  gas  ............................  0.558 

Carbonic  oxide  .........................  0.955 

A  second  compression  of  Tfy  of  the  new  volume  would,  as  we 
shall  soon  see,  again  raise  the  temperature  of  these  gases  In  an 
amount  nearly  equal  to  the  first  ;  but  this  would  not  be  the 
case  for  a  third,  a  fourth,  or  a  hundredth  compression  of  tin- 
same  sort.  The  capacity  of  gases  for  heat  changes  with  their 
volume;  it  is  quite  possible  that  it  changes  also  with  their 
temperature.* 

•  [It  »,u  found  ty  ReynauH  (M.-in.  •!.-  I'Acadfimle,  asnrf.,  p.  58)  tl. 
tprtifit  heat  of  the  "permanent"  gate*  it  independent  of  prtmure  and  tern- 
i  re.} 

'„'•! 


THE    SECOND   LAW    OF   THERMODYNAMICS 

We  shall  now  deduce  from  the  general  proposition  presented 
on  page  20  a  second  theorem  which  will  be  the  complement  of 
that  which  has  just  been  demonstrated. 

Let  us  suppose  that  the  gas  contained  in  the  cylinder  abed 
(Fig.  2)  is  transferred  to  the  receptacle  a'b'c'd'  (Fig.  3),  which  is 
of  equal  height,  but  which  has  a  different  and  larger  base  ;  the 
gas  will  increase  in  volume  and  diminish  in  density  and  elas- 
tic force  in  the  inverse  ratio  of  the  two  volumes  abed,  a'b'c'd'. 
The  total  pressure  exerted  on  each  piston,  cd,  c'd',  will  be  the 
same,  for  the  surfaces  of  these  pistons  are  in  the  direct  ratio  of 
the  volume. 

Let  us  suppose  that  the  operations  described  on  page  21  as 
performed  on  the  gas  contained  in  abed  are  performed  on  the 
gas  in  a'b'c'd' — that  is,  let  us  suppose  that  the  piston  c'd'  is  given 
displacements  equal  in  amplitude  to  those  given  the  piston  cd, 
and  that  it  occupies  successively  the  positions  c'd'  correspond- 
ing to  cd,  and  e'f  corresponding  to  ef.  At  the  same  time  let  us 
subject  the  gas,  by  means  of  the  two  bodies  A,  B,  to  the  same 
variations  of  temperature  as  those  to  which  it  was  subjected 
when  enclosed  in  abed ;  the  total  force  exerted  on  the  piston 
will  be  the  same  in  both  cases  at  corresponding  instants.  This 
results  immediately  from  Mariotte's  law*  ;  in  fact,  the  densities 
of  the  two  gases  are  always  in  the  same  ratio  for  similar  posi- 
tions of  the  pistons,  and,  their  temperatures  being  always  equal, 
the  total  pressures  exerted  on  the  pistons  are  always  in  the  same 
ratio.  If  this  ratio  is  at  any  time  that  of  equality,  the  pressures 
will  bo  always  equal. 

Further,  as  the  movements  of  the  two  pistons  have  equal  am- 
plitudes, the  motive  power  they  both  produce  will  evidently  be 
the  same,  from  which  we  may  conclude,  from  the  proposition 

*  Mariotte's  law,  upon  which  our  demonstration  is  based,  is  one  of  thebest- 
establishcd  physical  laws.  It  has  served  as  a  foundation  for  several  theo- 
ries verified  by  experiment,  and  which  verify  in  their  turn  the  laws  on 
which  they  rest.  We  may  also  cite,  as  an  important  verification  of 
Mariotte's  law  and  also  of  the  law  of  MM.  Gay-Lussac  and  Dalton  for  a 
large  range  of  temperature,  the  experiments  of  MM.  Dnlong  and  Petit. 
(See  Annales  de  Chimie  et  de  Physique,  Feb.,  1818,  vol.  vii.,  p.  122.)  We 
mav  also  cite  the  still  more  recent  experiments  of  Davy  and  Faraday. 

The  theorems  here  deduced  would  perhaps  not  be  exact  if  applied  out- 
side of  certain  limits  either  of  density  or  of  temperature.     They  should 
only  be  taken  as  true  within  the  limits  within  which  the  laws  of  Mariotte, 
Gay-Lussac,  and  Dalton  are  themselves  established. 
27 


MEMOIRS    ON 

on  page  20,  that  the  quantities  of  heat  used  by  each  are  equal — 
that  is  to  say,  that  the  same  quantity  of  heat  passes  from  .1  t<> 
//  in  each  case. 

The  heat  taken  from  the  body  A  and  given  to  the  body  /.'  i> 
nothing  other  than  the  heat  absorbed  by  the  expansion  of  the 
gas  and  afterwards  set  free  by  < •onipn-»iuii.  We  are  thus  led 
to  establish  the  following  theorem  : 

II  lit'ii  (lie  I'olnitli'  of  an  flitxtir  jlinil  rlminiis. without  r}/n)irjt<  nf 

tim/ierature,  from  I'  to  I".  "////  tin-  /•<<//////<  ,,/',/  i/nantitii  of  tin' 
same  gas,  equal  in  weight  and  at  tin'  >v////»-  fn,ij»  r,tf  >//••.  clntn<i>x 
from  IT1 toV',  the  quantities  of  heat  ali.^,rl,,;l  or  .^7  f !•<•>•  from  nn-li 
will  be  equal  when  the  ratio  of  U'  to  I  />•  «/mtl  to  tint!  of  U 
toV. 

This  theorem  may  be  stated  in  another  form,  as  follows: 

When  a  gas  changes  in  volume  without  change  of  fi-m/H-ratiin- 
the  quantities  of  heat  which  it  absorbs  or  gives  KI>  an-  in  arith- 
metical progression  when  the  increments  or  reductions  of  volume 
are  in  geometrical  /;/-o///v  »/'///. 

When  we  compress  one  litre  of  air  maintained  at  a  tempera- 
ture of  10  degrees  and  reduce  its  volume  to  J  a  litre,  it  gives 
out  a  certain  quantity  of  heat.  This  quantity  will  bo  al\\a\- 
the  same  if  we  further  reduce  the  volume  from  J  to  {,  from  J 
to  1 .  and  so  on. 

If,  instead  of  compressing  the  air,  we  allow  it  to  expand  to  2 
litres, 4  litres,8  litres, etc.,  successively,  we  must  supply  it  \\iih 
equal  quantities  of  heat  in  order  to  keep  its  tomporuturc  c«,nst  ant. 

This  easily  explains  why  the  temperature  of  air  risrs  \\-hrn  it 
is  suddenly  compressed.  We  know  that  this  temperature  is 
sufficient  to  ignite  tinder  and  even  to  cause  the  air  to  become 
luminous.  If  we  assume  for  the  time  lieing  the  specific  heat 
of  air  as  constant,  in  spite  of  changes  of  volume  and  tempera- 
ture, the  temperature  will  increase  in  arithmetical  pm^ressinn 
as  the  volume  is  diminished  in  geometrical  progression.  Start- 
ing with  this  as  given,  and  admitting  that  an  elevation  of  inn 
perature  of  1  degree  corresponds  to  a  compression  of  ^{r.  it  i> 
easy  to  conclude  that  when  air  is  reduced  to  fa  of  its  original 
volume  its  temperature  should  rise  about  300  degrees,  which 
is  enough  to  ignite  tinder.* 

•  When  the  volume  l»  mlur«<i  l,y  ,},— that  In.  when  it  becomes  \\\  of 
that  which  it  wa*  at  first— the  temperature  risen  1  <i< 


THE    SECOND    LAW    OF    THERMODYNAMICS 

The  elevation  of  temperature  would  evidently  be  still  greater 
if  the  capacity  of  the  air  for  heat  were  to  become  less  as  its 
volume  diminishes  ;  now  this  is  probable,  and  seems  to  be  con- 
firmed by  the  results  of  the  experiments  of  MM.  Delaroche  and 
Berard  on  the  specific  heat  of  air  taken  at  different  densities. 
(See  the  Memoir  published  in  the  Annalesde  Chimie  et  de  Phy- 
sique, vol.  Ixxxv.,  pp.  72,  224.) 

The  two  theorems  given  on  pages  23  and  28  are  sufficient  for 
the  comparison  of  the  quantities  of  heat  absorbed  or  released  in 
the  changes  of  volume  of  elastic  fluids,  whatever  may  be  the  den- 
sity and  chemical  nature  of  these  fluids,  provided  always  that 
they  are  taken  and  maintained  at  a  certain  invariable  tempera- 
ture ;  but  these  theorems  do  not  give  us  any  means  of  compar- 
ing quantities  of  heat  absorbed  or  released  by  elastic  fluids  whose 
volumes  are  changed  at  different  temperatures.  Thus  we  do 
not  know  the  relation  between  the  heat  released  by  1  litre  of  air 
reduced  in  volume  one-half  when  its  temperature  is  kept  at  zero 
and  the  heat  released  by  the  same  litre  of  air  reduced  in  volume 
one  -half  when  its  temperature  is  kept  at  100  degrees.  The 
knowledge  of  this  relation  is  connected  with  the  knowledge  of 
the  specific  heat  of  the  gases  at  different  degrees  of  tempera- 
ture, and  on  other  data  which  Physics,  in  its  present  state,  can- 
not furnish. 

The  second  of  our  theorems  affords  a  means  of  knowing  by 
what  law  the  specific  heat  of  gases  varies  with  their  density. 

Let  us  suppose  that  the  operations  described  on  page  21,  in- 
stead of  being  performed  with  two  bodies,  A,  B,  whose  temper- 


A  now  reduction  of  T|ff  brings  the  volume  to  (Ul)4.  and  tlie  tempera- 
ture should  rise  another  degree. 

After  x  such  reductions  the  volume  is  (Hf  )*.  aQd  the  temperature  should 
be  higher  by  x  degrees. 

If  we  set  (Hi^T1?.  and  tuke  tue  logarithms  of  both  sides,  we  find 
x  =  300°  about. 

If  we  set  (Hf)*  =  i  we  find  that  x  =  80°,  which  shows  that  the  tem- 
perature of  air  compressed  to  one  half  of  its  original  volume  rises  80  de- 


All  this  is  dependent  on  the  hypothesis  that  the  specific  heat  of  air  does 
not  change  when  the  volume  diminishes  ;  but  if,  for  the  reasons  given  on 
pages  31  and  32,  we  consider  the  specific  heat  of  air  compressed  to  one-half 
its  volume  as  reduced  in  the  ratio  of  700  to  616,  we  must  multiply  80  de- 
grees by  J^§,  which  brings  it  to  90  degrees. 
29 


MEMOIRS    ON 

atures  differ  by  an  infinitely  small  quantity,  are  performed  with 
two  bodies  whose  temperatures  differ  by  a  finite  quant ity,  say 
byl°. 

In  a  complete  cycle  of  operations  the  body  A  furnishes  to  the 
elastic  fluid  a  certain  quantity  of  heat  which  may  be  dividt •<! 
into  two  portions :  1,  the  quantity  required  to  keep  the  tem- 
perature of  the  fluid  constant  during  expansion  ;  2,  that  re- 
quired to  change  the  temperature  of  the  fluid  from  that  of  the 
body  B  to  that  of  the  body  A,  after  the  fluid  has  been  restored 
to  its  original  volume  and  is  put  in  contact  with  the  body  J. 
Let  us  call  the  first  of  these  quantities  a  and  the  second  b. 
The  total  caloric  furnished  by  the  body  A  will  be  expressed  by 
a  +  b. 

The  caloric  transmitted  by  the  fluid  to  the  body  B  may  also 
be  divided  into  two  parts ;  one  of  which,  b',  is  due  to  the  cooling 
of  the  gas  by  the  body  B,  the  other,  «',  is  that  released  by  the 
gas  during  the  reduction  of  its  volume.  The  sum  of  these  two 
quantities  is  a'-f  b' ',  this  should  be  equal  to  a  -f-  b,  for  after  a 
complete  cycle  of  operations  the  gas  returns  exactly  to  its 
original  state.*  It  must  have  given  up  all  the  caloric  with 
which  it  had  at  first  been  supplied.  We  then  have 

a+b=a'+b', 
or,  a  —  a'  =  b  —  b'. 

Now,  from  the  theorem  given  on  page  28,  the  quantities  a  and 
a'  are  independent  of  the  density  of  the  gas,  always  provided 
that  the  quantity  of  the  gas  by  weight  remains  the  -aim-  ami 
that  the  variations  of  volume  are  proportional  to  the  original 
volume.  The  difference  a  — a'  should  satisfy  the  same  con- 
ditions, and  consequently  also  the  difference  b'  —  b,  which  is 
equal  to  it.  But  b'  is  the  caloric  necessary  to  raise  the  temper- 
ature of  the  gas  contained  in  nhnl  one  degree  (Fig.  2);  b'  is  the 
caloric  released  by  the  gaa  when  it  is  enclosed  in  «/»/.  and  its 
temperature  falls  one  degree.  These  quantities  can  serve  as  a 
measure  of  the  specific  heats.  We  an-  thus  led  to  establish  the 
following  proposition : 

•  [The  M*  here  made  of  the  caloric  theory  titiata  the  demonttration  and 
Uttdt  to  erroneout  eondunont.} 

M 


THE    SECOND    LAW    OF    THERMODYNAMICS 

The  change  made  in  the  specific  heat  of  a  gas  in  consequence  of 
a  change  of  volume  depends  only  upon  the  relation  between  the 
original  volume  and  that  which  results  from  the  change — that  is 
to  say,  the  difference  between  the  specific  heats  does  not  depend 
on  the  absolute  magnitudes  of  the  volumes  but  on  their  ratio. 

This  proposition  may  be  stated  in  another  way,  namely  : 

When  the  volume  of  a  gas  increases  in  geometrical  progression 
its  specific  heat  increases  in  arithmetical  progression. 

Thus,  if  a  is  the  specific  heat  of  air  taken  at  a  given  den- 
sity, and  a  +  h  its  specific  heat  when  its  density  is  one-half  this, 
its  specific  heat  will  be  a+'2h  when  its  density  is  one-quarter 
this,  a+'3h  when  its  density  is  one-eighth  this,  and  so  on. 

•  The  specific  heats  are  here  referred  to  weight.  They  are 
supposed  to  be  taken  at  constant  volume  ;  but,  as  we  shall  see, 
they  would  follow  the  same  law  if  they  were  taken  under  con- 
stant pressure. 

In  fact,  to  what  cause  is  due  the  difference  between  the 
specific  heats  taken  at  constant  volume  and  under  constant 
pressure  ?  It  is  due  to  the  caloric  required  in  the  latter  case 
to  produce  the  increase  of  volume.  Now,  by  Mariotte's  law, 
the  increase  of  volume  of  a  gas,  for  a  given  change  of  tempera- 
ture, should  be  a  definite  fraction  of  the  original  volume,  which 
fraction  is  independent  of  the  pressure.  From  the  theorem 
given  on  page  28,if  the  ratio  between  the  original  volume  and  the 
changed  volume  is  given,  the  heat  required  to  produce  the  in- 
crease of  volume  is  determined  thereby.  It  depends  only  on  this 
ratio  and  on  the  quantity  of  the  gas  by  weight.  "We  must  then 
conclude  that :  The  difference  between  the  specific  heat  under  con- 
stant pressure  and  that  at  constant  volume  is  always  the  same, 
whatever  the  density  of  the  gas,  provided  that  the  quantity  of  the 
gas  by  weight  remains  the  same.  These  specific  heats  both  in- 
crease as  the  density  of  the  gas  diminishes,  but  their  difference 
does  not  change.*  Since  the  difference  between  the  two  capaci- 

*  MM.  Gay-Lussac  and  Welter  have  found  by  direct  experiments,  cited 
in  the  Mecanique  Celeste  and  in  the  Annales  de  Chimie  et  de  Physique,  July, 
1822,  page  267,  that  the  ratio  between  the  specific  heat  under  constant  press- 
ure and  that  at  constant  volume  varies  very  little  with  the  density  of  the 
gas.  From  what  we  have  just  seen,  it  is  the  difference  and  not  the  ratio 
that  should  remain  constant.  However,  as  the  specific  heats  of  gases,  for  a 
given  weight,  vary  very  little  with  their  density,  it  is  clear  that  the  ratio 
also  will  experience  only  very  small  changes. 
31 


M KM  Ml  US    ON 

ties  for  heat  is  constant,  when  one  increases  in  arithmetical 
progression  the  other  will  increase  in  a  similar  progression ;  thus 
our  law  applies  to  specific  heats  taken  under  constant  pressure. 

We  have  tacitly  supposed  that  the  specific  heat  inn. 
with  the  volume.  This  increase  is  shown  in  the  experiments 
of  MM.  Delaroche  and  Hi' rani;  these  physicists  have  found 
that  the  specific  heat  of  air  under  the  pressure  of  1  meter  of 
mercury  is  0.967  (see  the  memoir  already  referred  to),  taking  as 
the  unit  the  specific  heat  of  the  same  weight  of  air  under  tin- 
pressure  of  0.7GO  meter.  From  the  law  followed  hy  the  specific 
heats  with  respect  to  pressure,  observations  made  of  tln-m  in 
two  particular  cases  permit  us  to  calculate  them  in  all  pos- 
sible cases  ;  thus,  by  using  the  result  of  the  experiment  of  M  M. 
Delaroche  and  Berard,  which  has  just  been  cited,  we  have  con- 
structed the  following  table  of  the  specific  heats  of  air  under 
various  pressures : 


-PM'IFIC  HEAT, 

SPECIFIC  HEAT. 

PRESSURE 

THAT  OK  AIR  UNDER 

PKKSsritK 

THAT  OF  AIK   1  MU.Il 

IN 

ATMOSPHERIC 

IN 

ATMOSIMIKICK 

ATMOSPHERES 

PRESSURE 

\TMOH-IIKKI-.S 

PRESSURE 

BEING  1. 

•aura  i 

TiiW 

1.840 

1 

l.(MM) 

X 

1.756 

2 

0.916 

i 

L.678 

,688 

4 

8 

0.748 

Vr 

..~><>1 

16 

A 

.I'M 

.'  !  '.' 

0.580 

,886 

64 

0.496 

i 

.'.'•">'.' 

US 

0.411 

I 

'[.;.:, 

1 

.084 

51SJ 

ii  •>  |  j 

1 

.000 

M'.'l 

o!l60 

From  the  experiments  of  MM.  Gay-Lussac  and  WclU-r.  tlie  ratio  of  the 
specific  heat  of  atmospheric  air  under  constant  pressure  to  tint  ai  r«n 
stant  volume  is  1.8748,  a  uumtx-r  \\lii<  h  is  nearly  constant  for  all  pr. 
and  for  all  temperatures.     In  the  previous  iliHriivsion  we  have  l>c<>n  lid, 

by  other  considerations,  to  the  number  ~~na-j —  =1.44,  \\  hieli  dilTers  fnun 

this  by  fo,  and  we  have  used  this  number  to  construct  a  table  of  thesperiiic 
hents  of  gases  at  constant  volume.  Neither  this  taM.-  n«r  (lie  tai.l.-  uivrn 
on  page  88  should  lie  considered  as  accurate.  Tln-y  an-  intcmlr.1  mainly 
to  set  forth  the  laws  followed  by  the  specific  heats  of  aeriform  fluids. 


THE    SECOND    LAW    OF    THERMODYNAMICS 

The  numbers  in  the  first  column  are  in  geometrical  progres- 
sion, while  those  in  the  second  are  in  arithmetical  progression. 
We  have  carried  the  table  out  to  extreme  compressions  and 
rarefactions.  It  is  to  be  supposed  that  air,  before  attaining  a 
density  1024  times  its  ordinary  density  —  that  is,  before  becom- 
ing more  dense  than  water  —  would  be  liquefied.  The  specific 
heats  vanish,  and  even  become  negative  if  we  prolong  the 
table  beyond  the  last  number  given.  It  seems  probable  that  the 
numbers  in  the  second  column  decrease  too  rapidly.  The  ex- 
periments on  which  we  based  our  calculation  were  made  within 
too  narrow  limits  to  enable  us  to  expect  great  exactness  in  the 
numbers  obtained,  especially  in  the  extreme  values. 

Since,  on  the  one  hand,  we  know  the  law  by  which  heat  is 
evolved  by  the  compression  of  a  gas,  and,  on  the  other,  the  law 
by  which  the  specific  heat  varies  with  the  volume,  it  will  be 
easy  for  us  to  calculate  the  increase  of  temperature  of  a  gas 
compressed  without  loss  of  caloric.*  In  fact,  the  compression 
can  be  considered  as  consisting  of  two  successive  operations  : 
1,  compression  at  a  constant  temperature,  and,  2,  restoration 
of  the  caloric  emitted.  In  the  second  operation  the  tempera- 
ture will  rise  in  the  inverse  ratio  to  the  specific  heat  which 
the  gas  acquires  by  the  reduction  of  its  volume.  "We  can  de- 
termine the  specific  heat  by  means  of  the  law  above  demon- 
strated. From  the  theorem  on  page  28  the  heat  set  free  by 
compression  should  be  represented  by  an  expression  of  the  form 

s  =  A  +  B  log  v, 

s  being  the  heat,  v  the  volume  of  the  gas  after  compression, 
A  and  B  arbitrary  constants  dependent  on  the  original  volume 
of  the  gas,  on  its  pressure,  and  on  the  units  which  are  chosen. 

The  specific  heat,  which  varies  with  the  volume  in  accordance 
with  the  law  just  demonstrated,  should  be  represented  by  an 
expression  of  the  form, 


A'  and  B'  being  arbitrary  constants  different  from  A  and  B. 
The  increase  of  temperature  which  the  gas  receives  by  com- 

pression is  proportional  to  the  ratio  -,  or  to  the  ratio  -p  —       "    . 


*  [This  demonstration  is  erroneous  in  that  it  assumes  tJie,  materiality  of 
heat,  and  also  tfie  change  of  specific  heat  with  volume.     The  conclusions  are 

invalid.] 


MEMOIRS    ON 

It  may  be  represented  by  this  ratio  itself;  tlins,  if  we  represent 
it  by  /,  we  shall  have  A  +  li  log  r 

~  A  ~  /'  io 

If  the  original  volume  of  the  gas  is  1  and  the  original  u-nipt'ni- 
ture  zero,  we  shall  have  at  the  same  time  /  =  0,  log  y  =  0,  ami 
hence  A  —  0  ;  t  will  then  express  not  only  the  increase  of  tem- 
perature, but  the  temperature  itself  above  the  thermometric  zero. 
We  must  not  think  that  we  can  apply  the  formula  just  given 
to  very  large  changes  in  the  volume  of  the  gas.  We  have  taken 
the  rise  of  temperature  to  be  in  the  inverse  ratio  to  the  specific 
heat,  which  implies  that  the  specific  heat  is  constant  at  all  tem- 
peratures. Large  changes  of  volume  in  the  gas  occasion  large 
changes  of  temperature,  and  there  is  no  evidence  that  the  specific 
heat  is  constant  at  different  temperatures,  especially  when  these 
temperatures  are  widely  separated  from  each  other.  This  con- 
stancy of  specific  heat  is  only  an  hypothesis  assumed  in  the  case 
of  gases  from  analogy,  and  verified  fairly  well  for  solids  and  li(j  aids 
within  a  certain  range  of  the  thermometric  scale, but  which  the 
experiments  of  MM.  Pulongaml  Petit  have  shown  to  IK-  inexact 
when  extended  to  temperatures  much  above  100  degrees.* 

•  We  sec  no  reason  to  assume  a  priori  the  constancy  of  the  specific 
heat  of  bodies  at  various  temperatures  —  that  is  to  say,  to  assume  that 
equal  quantities  of  heat  will  produce  equal  increments  in  tin-  temperature 
of  a  body,  even  when  neither  the  state  in>r  tlic  density  of  the  body  U 
clinnired  ;  as.  for  example,  in  the  case  of  an  elastic  fluid  enclosed  in  n  riirul 
envelope.  Direct  experiments  on  solid  and  liquid  bodies  have  proved  that 
between  zero  and  100  degrees  equal  im -r. •im-nt*  of  heat  produce  nearly 
equal  increments  of  temperature ;  but  the  more  recent  experiments  of 
MM  Dulong  and  Petit  (see  AnnaU*  de  Cliimi,  ,t  <!•  l'/i?/*iqnf,  February. 
MUM -li.  and  April,  1818)  Imve  shown  that  this  relation  does  not  hold  for 
temperatures  much  over  100  degrees,  whether  they  are  measured  by  the 
mercury  thermometer  or  by  the  air  thermometer. 

Not  only  do  the  specific  heats  not  u-nwin  the  same  at  different  t.  in 
peratures.  but  the  ratios  between  them  do  not  remain  the  same  ;  so  Unit 
DO  thermometric  scale  can  establish  the  constancy  of  all  specific  heats  at 
the  same  time.  It  would  be  interesting  to  examine  whether  the  same 
irregularities  would  obtain  in  gaseous  substances,  but  the  necessary  experi- 
ments present  almost  insurmountable  ditllculiL-r 

It  teems  probable  that  the  irregularities  of  the  specific  beats  of  solid 
bodies  may  be  attributed  to  latent  heat,  employed  in  producing  a  commence- 
ment of  fusion,  a  softening  which  in  many  cases  becomes  perceptible  in 
these  bodies  long  before  complete  fusion  occurs.  We  can  support  this 
opinion  by  the  following  observation:  From  the  experiments  of  MM.  Dulong 
and  Petit,  the  increase  of  specific  hent  with  the  temperature  is  more  r.ij.i.i 
84 


THE  SECOND  LAW  OF  THERMODYNAMICS 

According  to  a  law  due  to  MM.  Clement  and  Desormes,* 
established  by  direct  experiment,  water  vapor,  under  whatever 
pressure  it  is  formed,  always  contains  the  same  quantity  of  heat 
in  equal  weights ;  this  amounts  to  saying  that  the  vapor  when 
compressed  or  expanded  without  loss  of  heat  is  always  in  such 
a  condition  as  to  saturate  the  space  which  it  occupies,  if  it  is 
originally  in  this  condition.  Water  vapor  in  this  condition  can 
thus  be  considered  a  permanent  gas,  and  should  follow  all  the 
laws  of  gases.  Consequently,  the  formula 

A  +  B\ogv 
~  A'+U'\ogv 

should  be  applicable  and  should  agree  with  the  table  of  tensions 
constructed  by  M.  Dalton  from  his  direct  experiments. 

We  can,  in  fact,  satisfy  ourselves  that  our  formula,  with  a 
suitable  determination  of  the  arbitrary  constants,  represents 
very  approximately  the  results  of  experiment.  The  unimpor- 
tant discrepancies  which  we  find  in  it  are  no  more  than  may 
reasonably  be  attributed  to  errors  of  observation.! 

with  solids  than  with  liquids,  though  the  latter  have  a  larger  dilatability. 
If  the  cause  which  we  have  proposed  to  account  for  this  irregularity  is  the 
real  one,  it  would  disappear  entirely  with  gases. 

*  [/it  has  been  shown  by  Rankine  and  Clausim  that  this  law  does  not  hold.] 
t  To  determine  the  arbitrary  constants  A,  B,  A,  B',  from  data  taken 
from  M.  Dalton's  table,  we  must  begin  by  calculating  the  volume  of  the 
vapor  by  means  of  its  pressure  and  temperature,  the  quantity  of  the  vapor 
by  weight  being  always  constant.  This  is  made  easy  by  the  laws  of  Ma- 
riotte  and  Gay-Lussac.  The  volume  will  be  given  by  the  equation 


in  which  v  is  the  volume,  t  the  temperature,  p  the  pressure,  and  c  a  constant 
quantity  which  depends  on  the  weight  of  the  vapor  and  the  units  chosen. 
The  following  is  the  table  of  the  volumes  occupied  by  a  gram  of  vapor 
formed  at  various  temperatures  and  consequently  under  various  pressures  : 


t 

p 

OR   THE  TENSION  OP  THE   VAPOR 

» 

OR  THE   VOLUME   OP    A    GRAM 

OR  CEOTIGRADE  DEGREES 

KXI'KKSSKl)   IN   MILLIMETRES 

OP   VAPOR   EXPRESSED 

OP   MERCURY 

IS  LITRES 

Deg. 

Mm. 

Lit. 

0 

5.060 

185.0 

20 

17.32 

58.2 

40 

53.00 

20.4 

60 

144.6 

7.96 

80 

352.1 

8.47 

100 

760.0 

1.70 

The  first  two  columns  in  this  table  are  taken  from  the  Traite  de  Physique 
85 


MKMOIRS    ON 

We  shall  now  retnrn  to  our  principal  subject,  the  motive 
power  of  heat,  from  which  we  have  already  digressed  too  far. 

We  have  shown  that  the  quantity  of  motive  power  developed 
by  the  transfer  of  caloric  from  one  body  to  another  depends 
essentially  on  the  temperatures  of  the  two  bodies,  but  we  have 
not  discussed  the  relation  between  these  temperatures  and  the 
quantities  of  motive  power  produced.  It  would  seem  at  tir.-t 
natural  enough  to  suppose  that  for  equal  differences  of  tem- 
perature the  quantities  of  motive  power  produced  are  equal — 
that  is,  for  example,  that  a  given  quantity  of  caloric  passing 
from  a  body,  A,  kept  at  100  degrees,  to  a  body,  B,  kept  at  50  de- 
grees would  develop  a  quantity  of  motive  power  equal  to  that 
which  would  be  developed  by  the  transfer  of  the  same  caloric 
from  a  body,  #,  kept  at  50  degrees  to  a  body,  C,  kept  at  zero. 
Such  a  law  would  indeed  be  a  very  remarkable  one,  hut  we.  do 
not  see  sufficient  reason  to  admit  it  a  priori.  We  shall  examine 
this  question  by  a  rigorous  method. 

Let  us  suppose  that  the  operations  described  on  page  21  are 
performed  successively  on  two  quantities  of  atmospheric  air 
equal  in  weight  and  volume  but  taken  at  different  temperat  :in •>. 
and  let  us  suppose  also  that  the  differences  of  temperature  l>e- 

of  M.  Biot  (vol.  i.,  pp.  272  mid  531).  The  third  is  calculated  by  menus  of 
the  above  formula,  mid  from  tin-  expuriincntul  fact  that  tin-  vapor  of  water 
under  atmospheric  pressure  occupies  a  volume  1700  times  us  gn -at  a>  ili.it 
which  it  occupies  when  in  the  liquid  state. 

By  using  three  numbers  from  the  first  column  and  the  corresponding 
numbers  from  the  third,  we  can  easily  determine  the  constants  of  our 
equation 

t  _  A  +  n  lotr  • 


logc 

We  shall  not  enter  into  the  dotnils  of  the  calculation  necessary  to  de- 
termine tin-si-  quantities;  it  will  be  enough  for  us  to  say  that  the  following 


B  =  -  1000.          &  =  8.80. 
satisfy  sufficiently  well  the  prescrilted  conditions,  so  that  the  equation 

=  19.04 

expresses  very  approximately  the  relation  existing  between  the  volume  of 
the  vnpor  and  its  temperature. 

It  is  to  be  noticed  that  the  quantity  IT  is  positive  and  very  small.  \\  hirli 
tends  to  confirm  the  proposition  tint  the  specific  heat  of  an  elastic  fluid  in- 
with  the  volume,  but  at  a  wry  slow  rate. 
80 


THE   SECOND   LAW   OF   THERMODYNAMICS 

tween  the  bodies  A  and  B  are  the  same  in  both  cases ;  thus, 
for  example,  the  temperatures  of  these  bodies  will  be  in  one 
case  100°  and  100°—  h  (h  being  infinitely  small),  and  in  the 
other,  1°  and  1°— h.  The  quantity  of  motive  power  produced 
is  in  each  case  the  difference  between  that  which  the  gas  fur- 
nishes by  its  expansion  and  that  which  must  be  used  to  restore 
it  to  its  original  volume.  Now  this  difference  is  here  the  same 
in  both  cases,  as  >ve  may  satisfy  ourselves  by  a  simple  argument, 
which  we  do  not  think  it  necessary  to  give  in  full ;  so  that  the 
motive  power  produced  is  the  same.  Let  us  now  compare  the 
quantities  of  heat  used  in  the  two  cases.  In  the  first  case  the 
quantity  used  is  that  which  the  body  A  imparts  to  the  air  in 
order  to  keep  it  at  a  temperature  of  100  degrees  during  its  ex- 
pansion ;  in  the  second,  it  is  that  which  the  same  body  imparts 
to  it  to  maintain  its  temperature  at  1  degree  during  an  exactly 
similar  change  of  volume.  If  these  two  quantities  were  equal 
it  is  evident  that  the  law  which  we  have  assumed  would  follow. 
But  there  is  nothing  to  prove  that  it  is  so ;  we  proceed  to  prove 
that  these  quantities  of  heat  are  unequal. 

The  air  which  we  first  supposed  to  occupy  the  space  abed 
(Fig.  2)  and  to  be  at  a  temperature  of  1  degree,  may  be  made  to 
occupy  the  space  abef,  and  to  acquire  the  temperature  of  100 
degrees  by  two  different  methods  : 

TL.  It  may  first  be  heated  without  change  of  volume,  and  then 
expanded  while  its  temperature  is  kept  constant. 

2.  It  may  first  be  expanded  while  its  temperature  is  kept 
constant,  and  then  heated  when  it  has  acquired  its  new  vol- 
ume. 

Let  a  and  b  be  the  quantities  of  heat  used  successively  in  the 
first  of  the  two  operations,  and  b'  and  «'  the  quantities  used  in 
the  second  ;  as  the  final  result  of  these  two  operations  is  the 
same,  the  quantities  of  heat  used  in  each  should  be  equal ;  we 
then  obtain 

from  which  we  have 

a'—a=b  —  b'. 

We  represent  by  a'  the  quantity  of  heat  necessary  to  raise  the 
temperature  of  the  gas  from  1  to  100  degrees  when  it  occupies 
the  volume  abef,  and  by  a  the  quantity  of  heat  necessary  to 
raise  the  temperature  of  the  gas  from  1  to  100  degrees  when  it 
occupies  the  volume  abed. 

37 


MEMOIRS    ON 

The  density  of  the  air  is  less  in  the  first  case  than  in  the  sec- 
ond, and  from  the  experiments  of  MM.  Delaroche  and  Beranl. 
already  cited  on  page  32,  its  capacity  for  heut  should  be  a  little 
greater. 

As  the  quantity  a'  is  greater  than  the  quantity  a,  b  should  be 
greater  than  b',  consequently,  stating  the  proposition  generally, 
we  may  say  that : 

The  quantity  of  hrat  <lnc  t<>  the  clntmjr  of  volume  of  a  gas  be- 
comes greater  as  the  teiiifn nttiirc  /">•  rai*nl. 

Thus,  for  example,  more  caloric  is  required  to  maintain  at 
100  degrees  the  temperature  of  a  certain  quantity  of  air  whose 
volume  is  doubled  than  to  maintain  at  1  degree  t)ie  tempera- 
ture of  the  same  quantity  of  air  during  a  similar  expansion. 

These  unequal  quantities  of  heat  will,  however,  as  we  have 
seen,  produce  equal  quantities  of  motive  power  for  equal  de- 
scents of  caloric  occurring  at  different  heights  on  the  thermo- 
metric  scale;  from  which  we  may  draw  the  following  conclu- 
sion : 

The  descent  of  caloric  prmlnn's  more  motive  power  at  lower  de- 
ffftU  of  temjH'raturr  limn  tit  higher.* 

Thus  a  given  quantity  of  heat  will  develop  more  motive 
power  in  passing  from  a  body  whose  temperature  is  kept  at  1 
degree  to  another  whose  temperature  is  kept  at  zero  than  if 
the  temperatures  of  these  two  bodies  had  been  lul  and  loo 
respectively.  It  must  be  said  that  the  difference  should  lie  very 
small  ;  it  would  be  zero  if  the  capacity  of  air  for  heat  remained 
constant  in  spite  of  changes  of  density.  According  to  the  ex- 
periments of  MM.  Delaroche  and  Beranl.  this  capacity  \arie> 
very  little,  so  little,  indeed,  that  the  differences  notiee<l  miu'ht 
strictly  be  attributed  to  errors  of  observation  or  to  s..me  eir- 
cumstances  which  were  not  taken  into  account. 

It  would  be  out  of  the  question  for  us,  with  the  experimental 
data  at  our  command,  to  determine  rigorously  the  law  l.y  which 

*  [  The  preff  fling  drmon*tration  if  erroMout  in  contequenfe  of  the.  attump- 
Hun  of  the  materiality  of  heat.     The  ennclwtion  it  inform  correct,  but  only 
becaute  of  the  erroneout  ute  of  a  variable  tpeciftc  heat  of  air.     If  tftit  be  con- 
tutored eonttant.  at  Carnot  point*  out.  the  efficiency  ihould  br.  <m  /•/»/,/•/. 
the  tame  at  all  temperaturet.     The  ratio  of  thr  n,>  ../.,•.•../  t->  th, 

heat  vted  »hould  be  equal  to  the  difference  of  tempernt'tr,  i,,<itti)>lied  by  a  eon- 
tfiint.  I/if  '•  C>i run ('»  fnitftiini."     At  «M  note  kiunr,  (hit  /  •/  Con- 

•t.ii.t.  but  it  the  reciprocal  of  the  abtolute  temperature  of  the  touree  of  htat.  ] 


THE   SECOND   LAW   OF   THERMODYNAMICS 

the  motive  power  of  heat  varies  at  different  degrees  of  the 
thermometric  scale.  It  is  connected  with  the  law  of  the  varia- 
tions of  the  specific  heat  of  gases  at  different  temperatures, 
which  has  not  been  determined  with  sufficient  exactness.*  We 


*  If  we  admit  that  the  specific  heat  of  a  gas  is  constant  when  its  volume 
docs  not  change,  but  only  its  temperature  varies,  analysis  would  lead  us  to 
a  relation  between  the  motive  power  and  the  therraometric  degree.  We 
shall  now  examine  the  way  in  which  this  may  be  done;  it  will  also  give  us 
an  opportunity  of  showing  how  some  of  the  propositions  formerly  estab- 
lished should  be  stated  in  algebraic  form. 

Let  /•  be  the  quantity  of  motive  power  produced  by  the  expansion  of  a 
given  quantity  of  air  changing  from  the  volume  1  litre  to  the  volume  v 
litres  at  constant  temperature.  If  v  increases  by  the  infinitely  small  quan- 
tity dv,  r  will  increase  by  the  quantity  dr,  which,  from  the  nature  of  mo- 
tive power,  will  be  equal  to  the  increase  of  volume  do  multiplied  by  the 
expansive  force  which  the  elastic  fluid  then  has.  If  p  represents  the  ex- 
pansive force,  we  shall  have  the  equation 

(1)  dr=pdv. 

Let  us  suppose  the  constant  temperature  at  which  the  expansion  occurs  to 
be  equal  to  t  degrees  centigrade.  Representing  by  q  the  elastic  force  of 
the  air  at  the  same  temperature,  t,  occupying  the  volume  of  1  litre,  we 
shall  have  from  Mariotte's  law 

—  =  —  ,  from  which  p  =  -- 

lp  v 

Now  if  Pis  the  elastic  force  of  the  same  air  always  occupying  the  volume 
1,  but  at  the  temperature  zero,  we  shall  have  from  M.  Gay-Lussac's  law 


T> 

If,  for  the  sake  of  brevity,  we  represent  by  N  the  quantity  ~,  the  equa- 
tion will  become 


by  using  which  we  have,  from  equation  (1), 


v 
Considering  t  constant,  and  taking  the  integrals  of  the  two  terms,  we  ob- 


If  we  suppose  that  r=Q  when  c=l,  we  shall  have  (7=0,  from  which 

(2)  r  =  ^V(<  +  267)logc. 

This  is  the  motive  power  produced  by  the  expansion  of  the  air  at  the  tem- 
perature t,  whose  volume  has  changed  from  1  to  «.     If  instead  of  working 


MHMOIRS    OX 

shall  now  endeavor  to  determine  definitively  the  motive  power 
of  heat,  and  in  order  to  verify  our  fundamental  proposition  — 

at  the  temperature  t  we  work  in  exactly  the  same  way  at  the  temperature 
t+dt,  the  power  developed  will  be 

r  +  £/•  =  y  (t  +  (It  +  2G7  1  log  r. 
Subtracting  equation  (2)  we  obtain 

(8)  Sr  =  N  log  oft. 

Let  •  be  the  quantity  of  heat  used  to  keep  the  temperature  of  the  gns 
constant  during  its  expansion.  From  the  discussion  on  page  21  ir  will  be 
the  power  developed  by  the  descent  of  the  quantity  of  heat  <  from  the  degree 
t  +  dt  to  the  degree  t.  Let  «  represent  the  motive  power  developed  by  the 
descent  of  a  unit  of  heat  from  t  degrees  to  zero;  since  from  the  gi-m  ml 
principle  established  ou  page  21  this  quantity  n  should  depend  only  on  t, 
it  may  be  represented  by  the  function  J-'t,  fn.ni  which  u  =  Ft. 

When  t  increases  and  becomes  i-\-<it.  n  becomes  v  +  du,  from  which 


Subtracting  the  preceding  equation  we  have 

du=F(t+dl)-Fl  =  Ftdt. 

This  is  evidently  the  quantity  of  motive  power  produced  by  the  descent  of 
a  unit  quantity  of  heal  from  the  degree  t  +  dt  to  the  degree  /. 

If  the  quantity  of  heat,  instead  of  being  a  unit,  had  been  <  .  the  motive 
power  produced  would  have  been 

(4)  edu  =  eF'tdt. 

But  edu  is  the  same  as  (r.  both  being  the  power  developed  by  the  descent  of 
the  quantity  of  heat  <  from  the  degree  t  +  dt  to  the  degree  t  ;  consequently, 

edu  =  Si; 
and,  from  equations  (3)  and  (4), 

eF'tdt  =  JVlog«//; 

or,  dividing  by  Ftdt,  and  representing  by  T  the  fraction  ^-  .  which  is  a 
function  of  t  only,  we  have 


The  equation  e  =  T  log  c 

is  the  analytical  expression  of  the  law  stated  on  page  28  ;  it  is  the  same  for 

all  gases,  since  the  laws  we  have  used  are  common  i-  all 

If  we  represent  by  *  the  quantity  of  heal  required  to  chant''1  the  volume 
of  the  ait  with  which  we  arc  working  from  1  to  r.  and  the  temper-it  ure  from 
zero  to  t,  the  difference  between  «  and  r  will  IK;  the  quantity  of  heat  required 
to  change  the  temperature  of  the  air,  while  its  volume  remains  1.  from  /<  m 
to  t.  This  quantity  depends  on  /  only.  It  will  be  some  function  of  /,  and 
we  shall  have,  if  we  call  it  U, 


If  wo  differentiate  this  equation  with  rc*pe<-t  to  t  only  anil  represent  l.\    /' 
and  I'  the  differential  coefficients  of  T  and  U,  it  will  become 

(5)  ^ 

40 


THE    SECOND    LAW    OF    THERMODYNAMICS 

that  is,  to  show  that  the  quantity  of  motive  power  produced 
is  really  independent  of  the  agent  used  —  we  shall  choose  sev- 

-37  is  nothing  other  than  the  specific  heat  of  the  gas  at  constant  volume, 
and  our  equation  (5)  is  the  analytical  expression  of  the  law  stated  on  page 
31. 

If  we  suppose  the  specific  heat  to  be  constant  at  all  temperatures—  an 

hypothesis  which  was  discussed  on  page  34  —  the  quantity  -r  will  be  inde- 

pendent of  t,  and,  to  satisfy  equation  (5)  for  two  particular  values  of  v, 
T  and  U'  must  also  be  independent  of  t  ;  we  shall  then  have  T'  =  C,  a  con- 
stunt  quantity.  Multiplying  T  and  C  by  dtaud  integrating  both  sides  we 
find 

T=Ct  +  Cl; 

but  as  T=jj^  we  have 

N 

T 

Multiplying  both  sides  by  dt  and  integrating  we  obtain 


or,  changing  the  arbitrary  constants,  and  remembering  that  Ft  is  zero 
when  t  =  0°,  we  have 

(6)  Pt  =  Alog(l  +  ^. 

The  nature  of  the  function  Ft  is  thus  determined,  and  may  serve  us  as  a 
means  of  calculating  the  motive  power  developed  by  any  descent  of  heat. 
But  this  last  conclusion  is  based  on  the  hypothesis  of  the  constancy  of  the 
specific  heat  of  a  gas  whose  volume  does  not  change—  an  hypothesis  which 
experiment  has  not  yet  sufficiently  verified.  Until  there  are  further  proofs 
of  its  validity  equation  (6)  can  only  be  admitted  for  a  small  part  of  the 
thermometric  scale. 

The  first  term  in  equation  (5)  represents,  as  we  have  said,  the  specific 
heat  of  the  air  occupying  the  volume  «.  Experiment  has  taught  us  that 
this  specific  heat  varies  only  slightly  in  spite  of  considerable  changes  of 
volume,  so  'that  the  coefficient  T'  of  log  v  must  be  a  very  small  quantity. 
If  we  assume  that  it  is  zero  and  multiply  the  equation  T'  =  Q  by  dt  and 
then  integrate,  we  have 

T=  C,  a  constant  quantity. 
But 


from  which 

f-M^. 

from  which  we  may  conclude  by  a  second  integration  that 
Ft  =  At  +  B. 
41 


MEMOIRS    ON 

eral  such  agents  —  atmospheric  air,  water  vapor,  and  alcohol 
vapor. 

Let  us  take  first  atmospheric  air.  The  operation  is  effected* 
according  to  the  method  indicated  on  page  21.  We  make  the 
following  hypotheses : 

The  air  is  taken  under  atmospheric  pressure ;  the  tempera- 
ture of  the  body  A  is  j^Vr  °f  a  degree  above  zero  and  that  of  the 
body  B  is  zero.  We  see  that  the  difference  is,  as  it  should  be, 
very  small.  The  increase  of.  the  volume  of  the  air  in  our  operu- 
tion  will  be  ^fa+yfa  of  the  original  volume  ;  this  is  a  very 
small  increase  considered  absolutely,  but  large  relatively  to  the 
difference  of  temperature  between  A  and  B. 

The  motive  power  developed  by  the  two  operations  described 
on  page  21  taken  together  will  be  very  nearly  proportional  to  the 
increase  of  volume  and  to  the  difference  bet ween  the  two  press- 
ures exerted  by  the  air  when  its  temperature  is  0.001°  and  zero. 

According  to  the  law  of  M.  Gay-Lussac,  this  difference  is 
i  g  7*0  OF  of  the  elastic  force  of  the  gas,  or  very  nearly  T jrVoT  °f 
the  atmospheric  pressure. 

The  pressure  of  the  atmosphere  is  equal  to  that  of  a  column 
of  water  10T<jfr  meters  high  ;  , , ,i0 0 0  of  this  pressure  is  equal  to 
that  of  a  water  column  8>^oirg  x  10.40  meters  in  height. 

As  for  the  increase  of  volume,  it  is,  by  hypothesis,  1 1»  + jii 
of  the  original  volume — that  is,  of  the  volume  occupied  by  1 
kilogram  of  air  at  zero,  which  is  equal  to  0.77  cubic  meters, 
if  we  take  into  account  the  specific  gravity  of  air;  thus  the 
product,  (Tl¥  +  lWO.T7II*1¥l<UO 


expresses  the  motive  power  developed.  This  power  is  here 
estimated  in  cubic  meters  of  water  raised  to  the  height  of  1 
meter. 

If  we  carry  out  the  multiplications  indicated,  we  find  for  the 
product  0.000000372. 

Let  us  now  try  to  determine  the  quantity  of  heat  used  to  ob- 
tain this  result—that  is,  the  quantity  transferred  from  the  body 
A  to  the  body  B.  The  body  A  furnishes  : 


0  when  t  =  0,  D  is  zero  ;  thus 

Ft 

tlint  is  to  nay.  the  motive  power  produced  is  exactly  proportional  to  the 
descent  of  the  caloric.  This  is  the  analytical  expression  of  the  statement 
made  on  page  88. 

42 


THE    SECOND    LAW    OF    THERMODYNAMICS 


1.  The  heat  required  to  raise  the  temperature  of  1  kilogram 
of  air  from  zero  to  0.001°. 

2.  The  quantity  required  to  maintain  the  temperature  of  the 
air  at  0.001°  when  it  undergoes  an  expansion  of 


The  first  of  these  quantities  of  heat  may  be  neglected,  as  it  is 
very  small  in  comparison  with  the  second,  which  is,  from  the 
discussion  on  page  24,  equal  to  that  required  to  raise  the  tem- 
perature of  1  kilogram  of  air  under  atmospheric  pressure  1 
degree. 

The  specific  heat  of  air  by  weight  is  0.267  that  of  water,  from 
the  experiments  of  MM.  Delaroche  and  Berard  on  the  specific 
heat  of  gases.  If,  then,  we  take  for  the  unit  of  heat  the  quantity 
required  to  raise  1  kilogram  of  water  1  degree,  the  quantity  re- 
quired to  raise  1  kilogram  of  air  1  degree  will  be  0.267.  Thus 
the  quantity  of  heat  furnished  by  the  body  A  is 

0.267  unit. 

This  quantity  of  heat  is  capable  of  producing  0.000000372  unit 
of  motive  power  by  its  descent  from  0.001  to  zero. 

For  a  descent  one  thousand  times  as  great,  or  of  one  degree, 
the  motive  power  will  be  very  nearly  one  thousand  times  as 
great  as  this,  or 

0.000372. 

Now  if,  instead  of  using  0.267  unit  of  heat,  we  use  1000  units, 
the  motive  power  produced  will  be  given  by  the  proportion 
^Z^^-IOJLQ,  from  which  -r  =  f£f  =  1.395  units. 

Thus  if  1000  units  of  heat  pass  from  a  body 
whose  temperature  is  kept  at  1  degree  to  another 
at  zero,  they  will  produce  by  their  action  on  air 
1.395  units  of  motive  power. 

We  shall  compare  this  result  with  that  which 
is  obtained  from  the  action  of  heat  on  water 
vapor. 

Let  us  suppose  that  1  kilogram  of  water  is 
contained  in  the  cylinder  abed  (Fig.  4)  between 
the  base  ab  and  the  piston  cd,  and  let  us  assume 
also  the  existence  of  two  bodies,  A,  B,  each 
maintained  at  a  constant  temperature,  that  of 
A  being  higher  than  that  of  B  by  a  very  small 
quantity.  We  shall  now  imagine  the  following 
operations  : 

43 


Fig 


MEMOIRS    ON 

1.  Contact  of  the  water  with  the  body  A,  change  of  the 
position  of  the  piston  from  cd  to  ef,  formation  of  vapor  at  the 
temperature  of  the  body  A  to  fill  the  vacuum  made  1>\  tilt- 
increase  of  the  volume.  We  shall  assume  the  volume  atn-t 
to  be  large  enough  to  contain  all  the  water  in  a  state  of 
vapor ; 

x'.  Removal  of  the  body  A,  contact  of  the  vapor  with  the 
body  B,  precipitation  of  a  part  of  this  vapor,  decrease  of  its 
elastic  force,  return  of  the  piston  from  efto  ab,  and  liquefaction 
of  the  rest  of  the  vapor  by  the  effect  of  the  pressure  combined 
with  the  contact  of  the  body  B; 

3.  Removal  of  the  body  B,  new  contact  of  the  water  with 
the  body  A,  return  of  the  water  to  the  temperature  of  this 
body,  a  repetition  of  the  first  operation,  and  so  on. 

The  quantity  of  motive  power  developed  in  a  complete  cy- 
cle of  operations  is  measured  by  the  product  of  the  volume 
of  the  vapor  multiplied  by  the  difference  between  its  tensions 
at  the  temperatures  of  the  body  A  and  of  the  body  B  respec- 
tively. 

The  heat  used — that  is,  that  transferred  from  the  body  A  to 
the  body  B — is  evidently  the  quantity  which  is  required  to 
transform  the  water  into  vapor,  always  neglecting  the  small 
quantity  necessary  to  restore  the  water  from  the  tempt -rature 
of  the  body  li  to  that  of  the  body  A. 

Let  us  suppose  that  the  temperature  of  the  body  A  is  100 
degrees  and  that  of  the  body  B  99  degrees.  From  1C.  Dahoifs 
table  the  difference  of  these  tensions  will  be  26  millimetres  of 
mercury  or  0.36  meter  of  water.  The  volume  occupied  by  tin- 
vapor  is  1700  that  of  the  water,  so  that,  if  we  use  l  kilo-ram, 
it  will  be  170Q  litres  or  1.700  cubic  meters.  Thus  the  motive 
power  developed  is 

1.700x0.36  =  0.611  unit 
of  the  sort  which  we  used  before. 

The  quantity  of  heat  used  is  the  quantity  required  to  trans- 
form the  water  into  vapor,  the  water  beini:  already  at  a  ti-m- 
|H-rature  of  100  degrees.  This  quantity  has  !><•.  u  d<  t<  rmiind 
I >y  experiment ;  it  has  been  found  equal  to  A.*><»  decrees,  or, 
-leaking  with  greater  precision,  to  550  of  our  units  of 
heat 

Thus  it. c,i  i  unit  of  motive  power  result  from  the  use  of  550 
units  of  heat. 

44 


THE    SECOND    LAW    OF    THERMODYNAMICS 

The  quantity  of  motive  power  produced  by  1000  units  of  heat 
will  be  given  by  the  proportion 

550        1000    .  ,  .  ,          611 

•  = ,  from  which  x  =  -—  =  1.112. 


0.611         x    '  550 

Thus  1000  units  of  heat  transferred  from  a  body  maintained 
at  100  degrees  to  one  maintained  at  99  degrees  will  produce 
1.112  units  of  motive  power  when  acting  on  the  water  vapor. 
The  number  1.112  differs  by  nearly  ^  from  1.395,  which  was  the 
number  previously  found  for  the  motive  power  developed  by 
1000  units  of  heat  acting  on  air  ;  but  we  must  remember  that 
in  that  case  the  temperature  of  the  bodies  A  and  B  were  1 
degree  and  zero,  while  in  this  case  they  are  100  and  99  degrees 
respectively.  The  difference,  is  indeed  the  same,  but  the  tem- 
peratures on  the  thermometric  scale  are  not  the  same.  In  order 
to  obtain  an  exact  pomparison  it  would  be  necessary  to  calculate 
the  motive  power  developed  by  the  vapor  formed  at  1  degree 
and  condensed  at  zero,  and  also  to  determine  the  quantity  of 
heat  contained  in  the  vapor  formed  at  1  degree.  The  law  of 
MM.  Clement  and  Desormes,  to  which  we  referred  on  page  35, 
gives  us  this  information.  The  heat  used  in  turning  water 
into  vapor  (chaleur  constituante)  is  always  the  same  at  whatever 
temperature  the  vaporization  occurs.  Therefore,  since  550 
degrees  of  heat  are  required  to  vaporize  the  water  at  the  tem- 
perature of  100  degrees,  we  must  have  550  4-100,  or  650  degrees, 
to  vaporize  the  same  weight  of  water  at  zero. 

By  using  the  data  thus  obtained,  and  reasoning  in  other 
respects  quite  in  the  same  way  as  we  did  when  the  water  was 
at  100  degrees,  we  readily  see  that  1.290  is  the  motive  power 
developed  by  1000  units  of  heat  acting  on  water  vapor  between 
the  temperatures  of  1  degree  and  zero. 

This  number  approaches  1.395  more  nearly  than  the  other. 

It  only  differs  by  y1^,  which  is  not  outside  the  limits  of  prob- 
able error,  considering  the  large  number  of  data  of  different 
sorts  which  we  have  found  it  necessary  to  use  in  making  this 
comparison.  Thus  our  fundamental  law  is  verified  in  a  par- 
ticular case.* 

*  In  a  memoir  of  M.  Petit  (Annales  de  Chimie  et  de  Physique,  July,  1818, 
page  294)  there  is  a  calculation  of  the  motive  power  of  heat  applied  to  air 
and  to  water  vapor.  The  results  of  this  calculation  are  much  to  the  advan- 
tage of  atmospheric  air ;  but  this  is  owing  to  a  very  inadequate  way  of 
considering  the  action  of  heat. 

45 


MEMOIRS   ON 

We  shall  now  examine  the  case  of  heat  acting  on  alcohol  vapor. 

The  method  used  in  this  case  is  exactly  the  same  as  in  the 
case  of  water  vapor,  but  the  data  are  different.  Pure  alcohol 
boils  under  ordinary  pressure  at  78.7°  centigrade.  According 
to  MM.  Delaroche  ami  P.t'-rard,  1  kilogram  of  this  substance 
absorbs  207  units  of  heat  when  transformed  into  vapor  at  this 
same  temperature,  78.7°. 

The  tension  of  alcohol  vapor  at  1  degree  below  its  boiling- 
point  is  diminished  by  .,'.,.  und  is  J.  less  than  atmospho  it- 
pressure  (this  is  at  least  the  result  of  the  experiments  of  M. 
Hi-tancour,  an  account  of  which  was  given  in  the  second  part 
of  M.  IVony's  An-h i/tff »/•<•  lliidrnnUque,  pages  180,  !!»">).* 

We  find,  by  use  of  these  data,  that  the  motive  ]u>\vrr  de- 
veloped, in  acting  on  1  kilogram  of  alcohol  at  the  temperatures 
77.7°  and  78.7°,  would  be  0.251  unit. 

This  results  from  the  use  of  207  units  of  heat.  For  1000 
units  we  must  set  the  proportion 

207        1000   . 
— - —  = ,  from  which  Z= 1.230. 

''.  J  '  1  *C 

This  number  is  a  little  greater  than  1.112,  resulting  from  the  use 
of  water  vapor  at  100 and  99  degrees  ;  but  if  \vi-  assume  the  water 
vapor  to  be  employed  at  78  and  77  degrees,  we  find,  by  the  law 
of  MM.  Cli'ment  and  Desormes,  1.212  for  the  motive  power 
produced  by  1000  units  of  heat.  As  we  see,  this  number  ap- 
proaches 1.230  very  nearly  ;  it  only  differs  from  it  by  -j^. 

*  M.  Dalton  thought  that  he  hiul  discovered  thnt  the  vapors  of  different, 
liquids  exhibited  equal  tensions  tit  temperatures  on  tlie  tin -rm.mn -trie  scale 
equally  distant  from  their  boiling-points ;  this  law  Is,  however,  not  rigor- 
ously, but  only  approximately,  correct.  The  same  is  true  of  tlie  law  of 
the  ratio  of  the  latent  heat  of  vapors  to  their  densities  (see  ex 
from  a  memoir  of  M.  C.  Despreiz.  Annale*  de  Chimit  et  de  Ptiyrique,  vol. 
xvi..  p.  105.  and  vol.  xxiv..  p.  828).  Questions  of  this  kind  are  closely  con- 
nected with  those  relating  to  the  motive  power  of  heat  Davy  and  Km  .<>.,  \ 
recently  tried  to  recognize  the  changes  of  tension  of  liquefied  gases  for 
small  changes  of  temperature,  after  having  made  excellent  experiments 
on  the  liquefaction  of  gnses  by  the  effect  of  a  considerable  pressure, 
had  In  view  the  use  of  new  liquids  in  the  production  of  motive  p«>\\,  t 
(see  Annale*  de  Chimie  et  fa  Physiqnt,  January.  1*24.  p.  80).  From  the 
theory  given  above  we  can  predict  that  the  use  of  these  liquids  presents  no 
advantage  for  the  economical  use  of  heat.  The  advantage  could  only  be 
realized  at  the  low  temperature  at  which  it  would  be  possible  to  work,  and 
by  the  use  of  sources  from  which,  for  this  reason,  it  would  become  pos- 
sible to  extract  caloric. 

46 


THE    SECOND    LAW    OF    THERMODYNAMICS 

We  should  have  liked  to  have  made  other  comparisons  of  this 
kind — for  example,  to  have  calculated  the  motive  power  de- 
veloped by  the  action  of  heat  on  solids  and  liquids,  by  the  freez- 
ing of  water,  etc. ;  but  in  the  present  state  of  Physics  we  are 
not  able  to  obtain  the  necessary  data.*  The  fundamental  law 
which  we  wish  to  confirm  seems,  however,  to  need  additional 
verifications  to  be  put  beyond  donbt;  it  is  based  upon  the  the- 
ory of  heat  as  it  is  at  present  established,  and,  it  must  be  con- 
fessed, this  does  not  appear  to  us  to  be  a  very  firm  foundation. 
New  experiments  alone  can  decide  this  question;  in  the  mean 
time  we  shall  occupy  ourselves  with  the  application  of  the  the- 
oretical ideas  above  stated,  and  shall  consider  them  as  correct 
in  the  examination  of  the  various  means  proposed  at  the  pres- 
ent time  to  realize  the  motive  power  of  heat. 

It  has  been  proposed  to  develop  motive  power  by  the  action 
of  heat  on  solid  bodies.  The  mode  of  procedure  which  most 
naturally  presents  itself  to  our  minds  is  to  firmly  fix  a  solid 
body — a  metallic  bar,  for  example — by  one  of  its  extremities, 
and  to  attach  the  other  extremity  to  a  movable  part  of  the 
machine ;  then  by  successive  heating  and  cooling  to  cause  the 
length  of  the  bar  to  vary,  and  thus  produce  some  movement. 
Let  us  endeavor  to  decide  if  this  mode  of  developing  motive 
power  can  be  advantageous.  We  have  shown  that  the  way  to 
get  the  best  results  in  the  production  of  motion  by  the  use  of 
heat  is  to  so  arrange  the  operations  that  all  the  changes  of  tem- 
perature which  occur  in  the  bodies  are  due  to  changes  of  vol- 
ume. The  more  nearly  this  condition  is  fulfilled  the  better  the 
heat  will  be  utilized.  Now,  by  proceeding  in  the  manner  just 
described,  we  are  far  from  fulfilling  this  condition  ;  no  change 
of  temperature  is  here  due  to  a  change  of  volume ;  but  the 
changes  are  all  due  to  the  contact  of  bodies  differently  heated, 
to  the  contact  of  the  metallic  bar  either  with  the  body  which 
furnishes  the  heat  or  with  the  body  which  absorbs  it. 

The  only  means  of  fulfilling  the  prescribed  condition  would 
be  to  act  on  the  solid  body  exactly  as  we  did  on  the  air  in  the 
operations  described  on  page  18,  but  for  this  we  must  be  able  to 
produce  considerable  changes  of  temperature  solely  by  the 
change  of  volume  of  the  solid  body,  if,  at  least,  we  desire  to 

*  The  data  lacking  are  the  expansive   force  acquired   by  solids  and 
liquids  for  u  given  increase  of  temperature,  and  the  quantity  of  heat  ab- 
sorbed or  emitted  during  changes  in  the  volume  of  these  bodies. 
47 


MKNh'IRS    ON 

use  considerable  descents  of  caloric.  Now  this  seems  to  be 
impracticable,  for  several  considerations  lead  ns  to  think  that 
the  changes  in  the  temperature  of  solids  or  liquids  by  compres- 
sion and  expansion  are  quite  small. 

1.  We  often  observe  in  engines  (in  heat-engines  particularly) 
solid  parts  which  are  subjected  to  very  considerable  forces,  some- 
times in  one  sense  and  sometimes  in  another,  and  although  those 
forces  are  sometimes  as  great  as  the  nature  of  the  substances 
employed  will  permit,  the  changes  in  temperature  are  scarcely 
perceptible. 

2.  In  the  process  of  striking  medals,  of  rolling  plates,  or  of 
drawing  wires,  metals  undergo  the  greatest  compressions  to 
which  we  can  subject  them  by  the  use  of  the  hardest  and  most 
resisting  materials.     Notwithstanding  this  the  rise  in  tempera- 
ture is  not  great,  for  if  it  were,  the  steel  tools  which  we  u.<«-  in 
these  operations  would  soon  lose  their  temper. 

3.  We  know  that  it  is  necessary  to  exert  a  very  great  force 
on  solids  and  liquids  to  produce  in  them  a  reduction  of  volume 
comparable  to  that  which  they  undergo   by  cooling  (for  ex- 
ample, by  a  cooling  from  100  degrees  to  zero).     Now,  cooling 
requires  a  greater  suppression  of  caloric  than  would  be  n><|iiiiv<l 
by  a  simple  reduction  of  volume.     If  this  reduction  were  pro- 
duced by  mechanical  means  the  heat  emitted  could  not  change 
the  temperature  of  the  body  as  many  degrees  as  the  cooling. 
It  would,  however,  require  the  use  of  a  force  which  would  cer- 
tainly be  very  considerable.     Since  solid  bodies  are  susceptible 
to  but  small  changes  of  temperature  by  changes  of  volume,  and 
since,  moreover,  the  condition  for  the  best  use  of  heat,  in  the  de- 
velopment of  motive  power  is  that  any  change  of  temperature 
should  be  due  to  a  change  of  volume,  solid  bodies  do  not  seem 
to  be  well  adapted  to  realize  this  power. 

This  is  equally  true  in  the  case  of  liquids ;  the  same  reasons 
could  be  given  for  rejecting  them.* 

We  shall  not  speak  hero  of  the  practical  difficulties,  which  are 
innumerable.  The  movements  produced  by  the  expansion  and 
compression  of  solids  or  liquids  can  only  be  very  small.  To 
extend  these  movements  we  should  be  forced  to  use  complicated 

*  The  recent  experiment*  of  M.  Oersted  on  the  compressibility  of  W.H.T 
hnvc  shown  that  for  a  pressure  of  5  atmospheres  the  temperature  of  the 
liquid  undergoes  no  perceptible  change.  (See  Annalet  d»  Chimie  et  de 
Phyrique.  February,  1828,  p.  192.) 

48 


THE    SECOND    LAW    OF    THERMODYNAMICS 

mechanisms  and  also  materials  of  the  greatest  strength  to  trans- 
mit enormous  pressures ;  and,  finally,  the  successive  operations 
conld  only  proceed  very  slowly  compared  with  those  of  the 
ordinary  heat-engine,  so  that  even  large  and  expensive  ma- 
chines would  produce  only  insignificant  results. 

Elastic  fluids,  gases,  or  vapors  are  the  instruments  peculiar- 
ly fitted  for  the  development  of  the  motive  power  of  heat ;  they 
unite  all  the  conditions  necessary  for  this  service  ;  they  may  be 
easily  compressed,  and  possess  the  property  of  almost  indefinite 
expansion  ;  changes  of  volume  occasion  in  them  great  changes 
of  temperature,  and  finally  they  are  very  mobile,  can  be  easily 
and  quickly  heated  and  cooled,  and  readily  transported  from 
one  place  to  another,  so  that  they  are  able  to  produce  rapidly 
the  effects  expected  of  them. 

We  can  easily  conceive  of  many  machines  fitted  for  the  de- 
velopment of  the  motive  power  of  heat  by  the  use  of  elastic 
fluids,  but  however  they  are  constructed  in  other  respects,  the 
following  conditions  must  not  be  lost  sight  of  : 

1.  The  temperature  of  the  fluid  should  first  be  raised  to  the 
highest  degree  possible,  in  order  to  obtain  a  great  descent  of 
caloric  and  consequently  a  great  production  of  motive  power. 

2.  For  the  same  reason  the  temperature  of  the  refrigerator 
should  be  as  low  as  possible. 

3.  The  operations  must  be  so  conducted  that  the  transfer  of 
the  elastic  fluid  from  the  highest  to   the  lowest  temperature 
should  be  due  to  an  increase  of  volume — that  is,  that  the  cool- 
ing of  the  gas  should  occur  spontaneously  by  the  effect  of  ex- 
pansion. 

The  limits  to  which  the  temperature  of  the  fluid  can  be 
raised  in  the  first  operation  are  determined  only  by  the  tem- 
perature of  combustion  ;  they  are  very  much  higher  than  ordi- 
nary temperatures.  The  limits  of  cooling  are  reached  in  the 
temperature  of  the  coldest  bodies  which  we  can  conveniently 
use  in  large  quantities  ;  the  body  most  used  for  this  purpose  is 
the  water  available  at  the  place  where  the  operation  is  car- 
ried on. 

As  to  the  third  condition,  it  introduces  difficulties  in  the 
realization  of  the  motive  power  of  heat,  when  the  object  is  to 
profit  by  great  differences  of  temperature,  that  is  to  utilize 
great  descents  of  caloric.  For  in  that  case  the  gas  must  change 
from  a  very  high  temperature  to  a  very  low  one,  by  expansion, 
D  49 


MEMOIRS    ON 

which  requires  a  great  change  of  volume  and  density.  To 
effect  this  the  gas  must  at  first  be  subjected  to  a  very  high 
pressure,  or  it  must  acquire  by  expansion  an  enormous  volume, 
either  of  which  conditions  is  difficult  to  realize.  Tin-  first 
necessitates  the  use  of  very  strong  vessels  to  contain  the  gas 
when  it  is  at  a  high  pressure  and  temperature;  the  second  re- 
quires the  use  of  vessels  of  a  very  large  size. 

In  fact,  these  are  the  principal  obstacles  in  the  way  of  profit- 
ably using  in  steam-engines  a  large  portion  of  the  motive  power 
of  heat.  We  are  of  necessity  limited  to  the  use  of  a  small  de- 
scent of  caloric,  although  the  combustion  of  coal  furnishes  us 
with  the  means  of  obtaining  a  very  great  one.  In  the  use  of 
steam-engines  the  elastic  fluid  is  rarely  developed  at  a  pressure 
higher  than  0  atmospheres,  which  pressure  corresponds  to  near- 
ly lf»0  degrees  centigrade,  and  condensation  is  rarely  effected  at 
a  temperature  much  below  40  degrees;  the  descent  of  caloric 
from  100  to  40  degrees  is  120  degrees,  while  we  can  obtain  by 
combustion  a  descent  of  from  1000  to  2000  degrees. 

To  conceive  of  this  better,  we  shall  recall  what  we  have  pre- 
viously called  the  descent  of  caloric  :  It  is  the  transfer  of  heat 
from  a  body,  A,  at  a  high  temperature  to  a  body,  11,  whose  tem- 
perature  is  lower.  We  say  that  the  descent  of  caloric  is  lun 
degrees  or  1000  degrees  when  the  difference  of  temperature 
between  the  bodies  J  ami  />'  is  100  or  1000  decrees.  In  a 
steam-engine  working  under  a  pressure  of  6  atmospheres  the 
temperature  of  the  boiler  is  100  degrees.  This  is  the  tempera- 
ture of  the  body  A  ;  it  is  maintained  by  contact  with  the  fur- 
nace at  a  constant  temperature  of  lH"  decree*,  ami  affords  a 
continual  supply  of  the  heat  necessary  to  the  formation  of 
steam. 

The  condenser  is  the  body  /?;  it  is  maintained  by  means  of  a 
current  of  cold  water  at  an  almost  constant  temperature  of  4<> 
degrees,  and  continually  absorbs  the  caloric  which  is  rarrie.l  t<> 
it  by  the  steam  from  the  body  A.  The  difference  of  tempera- 
ture  between  these  two  bodies  is  160—40,  or  120  degrees ;  it  is 
for  this  reason  that  we  say  that  the  descent  of  caloric  is  in  this 
case  120  doL' 

1     il  i<  capable  of  producing  by  combustion  a  higher  t« -m- 

peratlire  than    lIMM)    decrees,  ami    the    temperature,    of    the    e.uld 

water  which  we  ordinarily  use  is  about  1<>  decrees,  so  that  we  can 

easily  obtain  a  descent  of  caloric  of  1000  degrees,  of  which  only 

50 


THE    SECOND   LAW   OF   THERMODYNAMICS 

120  degrees  are  utilized  by  steam-engines,  and  even  these  120 
degrees  are  not  all  used  to  advantage  ;  there  are  always  con- 
siderable losses  due  to  useless  re-establishments  of  equilibrium 
in  the  caloric. 

It  is  now  easy  to  perceive  the  advantage  of  those  engines 
which  are  called  high-pressure  engines  over  those  in  which  the 
pressure  is  lower :  this  advantage  depends  essentially  upon  the 
power  of  utilizing  a  larger  descent  of  caloric.  The  steam  being 
produced  under  greater  pressure  is  also  at  a  higher  tempera- 
ture, and  as  the  temperature  of  condensation  is  always  nearly 
the  same  the  descent  of  caloric  is  evidently  greater. 

But  to  obtain  the  most  favorable  results  from  high-pressure 
engines  the  descent  of  caloric  must  be  used  to  the  greatest  ad- 
vantage. It  is  not  enough  that  the  steam  should  be  produced 
at  a  high  temperature,  but  it  is  also  necessary  that  it  should 
attain  a  sufficiently  low  temperature  by  its  expansion  alone. 
It  should  thus  be  the  characteristic  of  a  good  steam-engine  not 
only  that  it  uses  the  steam  under  high  pressure,  but  that  it  uses 
it  under  successive. pressures  which  are  very  variable,  very  dif- 
ferent from  each  other,  and  progressively  decreasing.* 

*This  principle,  which  is  the  real  basis  of  the  theory  of  the  steam-engine, 
has  been  developed  with  great  clearness  by  M.  Clement  in  a  memoir  pre- 
sented to  the  Academy  of  Sciences  a  few  years  ago.  This  memoir  has 
never  been  printed,  and  I  owe  my  acquaintance  with  it  to  the  kindness  of 
the  author.  In  it  not  only  is  this  principle  established,  but  applied  to  va- 
rious systems  of  engines  actually  in  use  ;  the  motive  power  of  each  is  cal- 
culated by  the  help  of  the  law  cited  on  p.  35  and  compared  with  the  results 
of  experiment.  This  principle  is  so  little  known  or  appreciated  that  Mr. 
Perkins,  the  well-known  London  mechanician,  recently  constructed  an 
engine  in  which  the  steam,  formed  under  a  pressure  of  35  atmospheres,  a 
pressure  never  before  utilized,  experienced  almost  HO  expansion,  as  we 
may  easily  be  convinced  by  the  slightest  knowledge  of  the  engine.  It  is 
composed  of  a  single  cylinder,  which  is  very  small,  and  at  each  stroke  is 
entirely  filled  with  steam  formed  under  a  pressure  of  35  atmospheres.  The 
steam  does  no  work  by  expansion,  for  there  is  no  room  for  the  expansion 
to  take  place  :  it  is  condensed  as  soon  as  it  passes  out  of  the  small  cylinder. 
It  acts  only  under  a  pressure  of  35  atmospheres,  and  not,  as  the  best  usage 
would  require,  under  progressively  decreasing  pressures.  This  engine  of 
Mr.  Perkins  does  not  realize  the  hopes  which  it  at  first  excited.  It  was 
claimed  that  the  economy  of  coal  in  this  machine  was  T9ff  greater  than  in 
the  best  machines  of  Watt,  and  that  it  also  possessed  other  advantages 
over  them.  (See  Annales  de  Ghimie  et  de  Physique,  April,  1823,  p.  429.) 
These  assertions  have  not  been  verified.  Mr.  Perkins's  engine  may  never- 
theless be  considered  a  valuable  invention  in  that  it  has  proved  it  to  be 
51 


MEMOIRS    ON 


In  order  to  show,  to  a  certain  extent,  a  /W^r/Wi  the  advan- 
tage of  high-pressure  engines,  let  us  assume  that  the  strain 
formed  under  atmospheric  pressure  is  contained  in  H  cylimln- 

feasiblc  to  use  steam  under  much  higher  pressures  than  ever  before,  and 
because  when  properly  modified  it  may  lead  us  to  really  useful  tesults. 

Wait,  to  whom  we  <>\v.-  almost  nil  the  great  improvements  in  the  strum 
engine,  and  who  has  brought  these  machines  to  a  state  of  perfection  which 
can  hardly  IK?  surpassed,  was  the  first  to  use  steam  under  pr.>i:i-e**ivrly 
decreasing  pressures.  In  many  cases  he  checked  the  introduction  of  the 
steam  into  the  cylinder  at  one-half,  one-third,  or  one-quarter  of  tin-  stroke 
of  the  piston,  which  was  thus  completed  under  a  picssiirc  which  constant 
ly  diminished.  The  first  engines  working  on  this  principle  dale  from  177s 
Walt  had  conceived  the  idea  in  1769,  and  took  out  a  patent  for  it  in  17^,' 

A  table  annexed  to  Watt's  patent  is  here  presented.  In  it  lie  supposes 
the  vapor  to  enter  the  cylinder  during  the  first  quarter  of  the  stroke  of  the 
piston,  and  he  then  calculates  the  mean  pressure  by  dividing  the  stroke  in;.  . 
twenty  parts: 


l-ARTW   Or   TIIK 

PATH    FRO*  TUB  HBAD  OF  THK 
CYLINDER 

DKRKA8IKO   PRKMfRK  Or  TlUt  VAPOK,   TH« 
TOTAL    PMK881-HK   RUM.    1 

0.05  \ 
0.101    Steam  entering 

i)  i:>          freely  from 

,1.000} 

\1   no,,/ 

1.  000  V  Total  pressure. 

o-jo 

the  boiler. 

M.  -00, 

Quarter  

.0.25 

;  i.ooo  ) 

0.80 

0.810 

085 

0.714 

0.40 

0625 

0.45 

o  560 

Half.  

.0.50 

' 

0.500.  .Half  the  original  pressure. 

• 

ii  V, 
060 
0.65 
0.70 

a.  78 

The  steam  cut  off. 
and  moving  the 
piston  by  expan- 
sion aloue. 

o  »:,» 
0417 
0.885 
0875 

0.888..  One  third. 

OSO 

0.811 

0.85 

o  IM 

0.90 

Bottom  of 

095 

cylinder 

.1.00 

0.250.  .One-quarter. 

Total  

11  588 

Mean  pressure,  11'888  = 

0.579. 

On  this  showing  he  remarks  that  the  mean  pressure  is  more  thnn  half  of 
the  original  pressure,  so  that  a  quantity  of  steam  equal  t»  »ne  quarter  \\  dl 
produce  an  effect  greater  than  one-half  [freely  intrmlncctl  fmm  tin  hnlrr 
until  the  end  of  the  ftrolce}. 

Watt  here  assumes  that  the  i  \p:m-i»n  <>f  the  steam  is  in  accordance  with 
Mariotte's  law.  This  assumption  should  not  be  considered  correct,  be- 


THE   SECOND    LAW    OF   THERMODYNAMICS 

cal  vessel,  abed  (Fig.  5).  under  the  piston  cd,  which  at  first 

touches  the  base  ab;  the  steam,  after  moving  the  piston  from 

ab  to  cd,  will  subsequently  act  in  a  manner  with 

which  we  need  not  occupy  ourselves.     Let  us 

suppose  that  after  the  piston  has  reached  cd  it  is 

forced  down  to  ef  without  escape  of  steam,  or 

loss  of  any  of  its  caloric.     It  will  be  compressed 

into  the  space  abef,  and  its  density,  elastic  force, 

and  temperature  will  all  increase  together. 

If  the  steam,  instead  of  being  formed  under 
atmospheric  pressure,  were  produced  exactly  in 
the  state  in  which  it  is  when  compressed  into   a 
abef,  and  if,  after  having  moved  the  piston  from  #%.  5 

ab  to  ef  by  its  introduction  into  the  cylinder,  it 
should  move  it  from  ef  to  cd  solely  by  expansion,- the  motive 
power  produced  would  be  greater  than  in  the  first  case.  In 
fact,  an  equal  movement  of  the  piston  would  take  place  under 
the  influence  of  a  higher  pressure,  although  this  would  be  va- 
riable and  even  progressively  decreasing. 

The  steam  would  require  for  its  formation  a  precisely  equal 
quantity  of  caloric,  but  this  caloric  would  be  used  at  a  higher 
temperature. 

It  is  from  considerations  of  this  kind  that  engines  with  two 
cylinders  (compound  engines)  were  introduced,  which  were  in- 
vented by  Mr.  Hornblower  and  improved  by  Mr.  Woolf.  With 

cause,  on  the  one  hand  the  temperature  of  the  elastic  fluid  is  lowered  by  ex- 
pansion, and  on  the  other  there  is  nothing  to  show  that  a  part  of  this  fluid 
does  not  condense  by  expansion.  Watt  should  also  have  taken  into  ac- 
count the  force  necessary  to  expel  the  steam  remaining  after  condensa- 
tion, whose  quantity  is  greater  in  proportion  as  the  expansion  has  been 
carried  further.  Dr.  Robinson  added  to  Watt's  work  a  simple  formula 
to  calculate  the  effect  of  the  expansion  of  steam,  but  this  formula  is  af- 
fected by  the  same  errors  to  which  we  have  just  called  attention.  It  has, 
however,  been  useful  to  constructors  in  furnishing  them  with  a  means  of 
calculation  sufficiently  exact  to  be  of  use  in  practice  WTe  have  thought  it 
worth  while  to  recall  these  facts  because  they  are  little  known,  especially 
in  France.  Engines  have  been  constructed  there  after  the  models  of  in- 
ventors but  without  much  appreciation  of  the  principles  on  which  these 
models  were  made.  The  neglect  of  these  principles  has  often  led  to  grave 
faults.  Engines  which  were  originally  well  conceived  have  deteriorated 
in  the  hands  of  unskilful  constructors,  who,  wishing  to  introduce  unim- 
portant improvements,  have  neglected  fundamental  considerations  which 
they  did  not  know  enough  to  appreciate. 

53 


MEMOIRS    ON 

respect  to  the  economy  of  fuel,  they  are  considered  the  best  en- 
gines. They  are  composed  of  a  small  cylinder,  which  at  each 
stroke  of  the  piston  is  more  or  less  and  often  entirely  tilled 
with  steam,  and  of  a  second  cylinder,  of  a  capacity  usually 
fonr  times  as  great,  which  receives  only  the  steam  which  has 
already  been  used  in  the  first  one.  Thus  the  volume  of  the 
steam  at  the  end  of  this  operation  is  at  least  four  times  its 
original  volume.  It  is  carried  from  the  second  cylinder  direct- 
ly into  the  condenser;  but  it  is  evident  that  it  could  he  carried 
into  a  third  cylinder  four  times  as  large  as  the  second,  where  its 
volume  would  become  sixteen  times  its  original  volume.  The 
chief  obstacle  to  the  use  of  a  third  cylinder  of  this  kind  is  tin- 
large  space  which  it  requires,  and  the  size  of  the  openings 
which  are  necessary  to  allow  the  steam  to  escape.* 

We  shaU  say  nothing  more  on  this  subject,  our  object  ii<>t 
being  to  discuss  the  details  of  construction  of  heat-engines. 
These  should  be  treated  in  a  separate  work.  No  such  work 
exists  at  present,  at  least  in  France,  f 

*  It  is  easy  to  perceive  the  advantages  of  having  two  cylinders,  for  when 
there  is  only  one  the  pressure  on  the  piston  will  vary  very  much  he!  ween  the 
beginning  and  end  of  the  stroke.  Also,  all  the  portions  of  the  machine  de 
signed  to  transmit  the  action  must  be  strong  enough  to  re-ist  tin-  first  im- 
pulse, and  filled  together  pcifectly  so  as  to  avoid  sudden  motions  by  \\liii-h 
they  might  be  damaged  and  which  would  soon  wear  them  out.  Tins  \\onld 
be  specially  true  of  the  walking  beam,  the  supports,  tin-  conm< -ting-rod, 
the  crank,  and  of  the  first  cog-wheels.  In  these  parts  tin-  irreirnlarity  of 
the  impulse  would  be  roost  felt  and  would  do  tin-  most  damage  The 
steam-chest  would  also  have  to  lie  strong  enough  to  resist  tin-  i 
pressure  employed,  and  large  enough  to  contain  the  v:i|>or  when  its  volume 
is  increased.  If  two  cylinders  are  used  the  capacity  of  the  first  need  not 
be  great,  so  that  it  is  easy  to  give  it  the  strength  required,  while  the  second 
must  be  large  but  need  not  be  very  strong. 

Engines  with  two  cylinders  have  been  planned  on  proper  principles  hut 
have  often  fallen  far  short  of  yielding  a«  good  iwulis  n.s  miL'lit  have  been 
expected  of  them.  Thi-  is  the  case  principally  IK-CM u*e  the  dimer>- 
the  different  parts  are  difficult  to  arrange  and  are  often  not  in  proper  pro- 
portion to  each  other.  There  are  no  good  models  of  the-e  engines,  while 
there  are  excellent  ones  of  those  constructed  after  Watt's  plan  To  this  is 
due  the  it  regularity  which  we  observe  in  the  effects  produced  by  the 
former.  whil«  those  produced  by  the  latter  are  almost  uniform. 

4  In  the  work  entitled  De  la  fifc*MM  Vintral.  by  M.  Heron  de  V ill, .fosse, 
vol.  iii ,  p.  .V)  itq.,  we  find  a  good  description  of  the  steam  ermine-  M..W 
used  in  mining.     The  subject  Ir.s  b.  en  treated  wilh  sufficient   ful 
England  in  the  Kneyflnptnlia  Brittinnica.    Some  of  the  data  which  we 
have  employed  have  been  taken  from  the  latter  work. 
54 


THE    SECOND    LAW    OF    THERMODYNAMICS 

"While  the  expansion  of  the  steam  is  limited  by  the  dimen- 
sions of  the  vessels  in  which  it  dilates,  the  degree  of  conden- 
sation at  which  it  is  possible  to  begin  to  use  it  is  only  limited 
by  the  resistance  of  the  vessels  in  which  it  is  generated — name- 
ly, of  the  boilers.  In  this  respect  we  are  far  from  having 
reached  the  possible  limits.  The  character  of  the  boilers  in 
general  use  is  altogether  bad;  although  the  tension  of  the  steam 
is  rarely  carried  in  them  beyond  4  to  6  atmospheres,  they  often 
burst  and  have  caused  serious  accidents.  It  is  no  doubt  quite 
possible  to  avoid  such  accidents  and  at  the  same  time  to  make 
the  tension  of  the  steam  greater  than  that  commonly  employed. 

Besides  the  high  -  pressure  engines  with  two  cylinders  of 
which  we  have  been  speaking,  there  are  also  high-pressure  en- 
gines with  one  cylinder.  Most  of  these  have  been  constructed 
by  two  skilful  English  engineers,  Messrs.  Trevithick  and  Viv- 
ian. They  use  the  steam  under  a  very  high  pressure,  some- 
times 8  or  10  atmospheres,  but  they  have  no  condenser.  The 
steam,  after  its  entrance  into  the  cylinder,  undergoes  a  certain 
expansion,  but  its  pressure  is  always  greater  than  that  of  the 
atmosphere.  When  it  has  done  its  work,  it  is  ejected  into  the 
atmosphere.  It  is  evident  that  this  mode  of  procedure  is  en- 
tirely equivalent,  with  respect  to  the  motive  power  produced, 
to  condensing  the  steam  at  100  degrees,  and  that  we  lose  a  part 
of  the  useful  effect,  but  engines  thus  worked  can  dispense  with 
the  condenser  and  air-pump.  They  are  less  expensive  than  the 
others,  and  are  not  so  complicated ;  they  take  less  room,  and  can 
be  used  where  it  is  not  possible  to  obtain  :i  current  of  cold  water 
sufficient  to  effect  condensation.  In  such  places  they  possess 
an  incalculable  advantage,  since  no  others  can  be  used.  They 
are  used  principally  in  England  to  draw  wagons  for  the  carriage 
of  coal  on  railroads,  either  in  the  interior  of  mines  or  on  the 
surface. 

Some  remarks  may  still  be  made  on  the  use  of  permanent 
gases  and  vapors  other  than  water  vapor  in  the  development  of 
the  motive  power  of  heat. 

Various  attempts  have  been  made  to  produce  motive  power 
by  the  action  of  heat  on  atmospheric  air.  This  gas,  in  com- 
parison with  water  vapor,  presents  some  advantages  and  some 
disadvantages,  which  we  shall  now  examine. 

1.  It  has  this  notable  advantage  over  water  vapor,  that 
since  for  the  same  volume  it  has  a  much  smaller  capacity  for 
55 


MEMOIRS    <>N 

heat  it  cools  more  for  an  equal  expansion,  as  is  proved  by  what 
we  have  previously  said.  We  have  seen  the  importance  of  effect- 
ing the  greatest  possible  changes  of  temperature  by  changes  of 
volume  alone. 

2.  Water  vapor  can  be  formed  only  by  the  aid  of  a  boiler, 
while  atmospheric  air  can  be  heated  directly  by  combustion 
within  itself.     Thus  a  considerable  loss  is  avoided,  not  only  in 
the  quantity  of  heat,  but  also  in  its  thermometric  degree.    This 
advantage  belongs  exclusively  to  atmospheric  air ;  the  other 
gases  do  not  possess  it;  they  would  be  even  more  difficult  to 
heat  than  water  vapor. 

3.  In  order  to  produce  a  great  expansion  of  the  air,  and  to 
cause  thereby  a  great  change  of  temperature,  it  would  be  neces- 
sary to  subject  it  in  the  first  place  to  rather  a  high  pressure,  to 
compress  it  by  an  air-pump  or  by  some  other  means  before  heat- 
ing  it.     This  operation  would  require  a  special  apparatus  which 
does  not  form  a  part  of  the  steam-engine.     In  it  the  water  is 
in  a  liquid  state  when  it  enters  the  boiler,  and  requires  only  a 
small  force-pump  to  introduce  it. 

4.  The  cooling  of  the  vapor  by  the  contact  of  the  refrigerat- 
ing body  is  more  rapid  and  easy  than  the  cooling  of  air  could 
be.     It  is  true  that  we  have  the  resource  of  eject inn  it  into  the. 
atmosphere.     This  procedure  would   have  the  further  advan- 
tage that  we  could  then  dispense  with  a  refrigerator,  which  is 
not  always  at  our  disposal,  but  in  that  case  the  air  must   n»t 
expand  so  far  that  its  pressure  is  lower  than  that  of  the  a; 
phere. 

5.  One  of  the  most  serious  drawbacks  to  the  employment  of 
steam  is  that  it  cannot  be  used  at  high  temperatures  except 
with  vessels  of  extraordinary  strength.     This  is  not  true  <>f  air. 
for  which  there  is  no  necessary  relation  between  its  tempera- 
ture and  elastic  force.     The  air,  then,  seems  bettor  fitted  than 
steam  to  realize  the  motive  power  of  the  descent  of  caloric  at 
high  temperatures;  perhaps  at  low  temperatures  water  vapor 
would  be  preferable.     We  can  even  conceive  of  the  possibility 
of  making  the  same  heat  act  successively  in  air  and  in  water 
vapor.     All  that  would  be  necessary  would  be  to  keep  the  tem- 
perature of  the  air  sufficiently  high,  after  it  had  Wn  nsi-d.  and 
instead  of  ejecting  it  immediately  into  the  atmosphere,  to  sur- 
round a  steam-boiler  with  it,  as  if  it  came  directly  from  the 
fire-box. 

N 


THE    SECOND    LAW    OF    THERMODYNAMICS 

The  use  of  atmospheric  air  for  the  development  of  the  mo- 
tive power  of  heat  presents  very  great  practical  difficulties  which, 
however,  may  not  be  insurmountable.  These  difficulties  once 
overcome,  it  will  doubtless  be  far  superior  to  water  vapor.* 

As  for  other  permanent  gases,  they  should  be  finally  rejected ; 
they  have  all  the  inconveniences  of  atmospheric  air  without 
any  of  its  advantages. 

The  same  may  be  said  of  other  vapors  in  comparison  with 
water  vapor. 

*  Among  the  attempts  made  to  develop  the  motive  power  of  heat  by  the 
use  of  atmospheric  air,  we  should  notice  particularly  those  of  MM.  Niepce, 
which  were  made  in  France  several  years  ago  by  means  of  an  apparatus, 
called  by  the  inventors  pyreolophore.  This  instrument  consists  essentially 
of  a  cylinder,  furnished  with  a  piston,  and  tilled  with  atmospheric  air  at 
ordinary  density.  Into  this  is  projected  some  combustible  substance  in  a 
highly  attenuated  form,  which  remains  in  suspension  for  a  moment  in  the 
air  and  is  then  ignited.  The  combustion  produces  nearly  the  same  effect 
as  if  the  elastic  fluid  were  a  mixture  of  air  and  combustible  gas — of  air 
and  carburetted  hydrogen,  for  example — a  sort  of  explosion  occurs  and  a 
sudden  expansion  of  the  elastic  fluid,  which  is  made  use  of  by  causing  it 
to  act  altogether  against  the  piston.  This  moves  through  a  certain  dis- 
tance, and  the  motive  power  is  thus  realized.  There  is  nothing  to  prevent 
a  renewal  of  the  air  and  a  repetition  of  the  first  operation.  This  very  in- 
genious engine,  which  is  especially  interesting  on  account  of  the  novelty 
of  its  principle,  fails  in  an  esseniial  particular.  The  substance  used  for  the 
combustible  (lycopodium  powder,  that  which  is  used  to  produce  flames  on 
the  stage)  is  so  expensive,  that  all  other  advantages  are  outweighed,  and 
unfortunately  it  is  difficult  to  make  use  of  a  moderately  cheap  combustible, 
for  it  requires  a  substance  that  is  very  finely  pulverized,  in  which  the  igni- 
tion is  prompt,  is  propngnted  rapidly,  and  which  leaves  little  or  no  residue. 

Instead  of  following  MM.  Niepce's  operations  it  would  seem  to  us  better 
to  compress  the  air  by  air-pumps  and  to  conduct  it  through  a  perfectly 
sealed  fire-box  into  which  the  combustible  is  introduced  in  small  quan- 
tities by  some  mechanism  which  is  easy  to  conceive  of;  to  allow  it  to  de- 
velop its  action  in  a  cylinder  with  a  piston  or  in  any  other  envelope  capable 
of  enlargement ;  to  eject  it  finally  into  the  atmosphere,  or  even  to  pass  it 
under  a  steam-boiler  in  order  to  utilize  its  remaining  heat. 

The  chief  difficulties  which  we  should  have  to  meet  in  this  mode  of 
operation  would  be  the  enclosure  of  the  fire-box  in  a  sufficiently  solid  en- 
velope, the  suitable  control  of  the  combustion,  the  maintenance  of  a  moder- 
ate temperature  in  the  several  parts  of  the  engine,  and  the  prevention  of  O; 
rapid  deterioration  of  the  cylinder  and  piston.  We  do  not  consider  these 
difficulties  insurmountable. 

It  is  said  that  successful  attempts  have  been  made  in  England  to  develop 
motive  power  by  the  action  of  heat  on  atmospheric  air.     We  do  not  know 
what  these  are,  if,  indeed,  they  have  really  been  made. 
57 


MEMOIRS    ON 

It  would  no  doubt  be  preferable  if  there  were  an  abundant 
supply  of  a  liquid  which  evaporated  at  a  higher  temperature 
than  water,  the  specific  heat  of  whose  vapor  was  less  for  equal 
volume,  and  which  did  not  injure  the  metals  used  in  the  con- 
struction of  an  engine  ;  but  no  such  body  exists  in  natim-. 

The  use  of  alcohol  vapor  has  been  suggested,  and  engines 
have  even  been  constructed  in  order  to  make  it  possible,  in 
\\hii-li  the  mixture  of  the  vapor  with  the  water  of  condensation 
is  avoided  by  applying  the  cold  body  externally  instead  of  in- 
troducing it  into  the  engine. 

It  was  thought  that  alcohol  vapor  possessed  a  notable  advan- 
tage on  account  of  its  having  a  greater  tension  than  that  of 
water  vapor  at  the  same  temperature.  We  see  in  this  only  an- 
other difficulty  to  be  overcome.  The  principal  defect  of  water 
vapor  is  its  excessive  tension  at  a  high  temperature,  and  this 
defect  is  still  more  marked  in  alcohol  vapor.  As  for  the  ad- 
vantage which  it  was  believed  to  have  with  respect  to  a  larger 
production  of  motive  power,  we  know  from  the  principles  stated 
above  that  they  are  imaginary. 

Thus  it  is  with  the  use  of  water  vapor  and  atmospheric  air 
that  the  future  attempts  to  improve  the  steam-engine  shun  1.1 
be  made.  AH  efforts  should  be  directed  to  utilize  by  means  of 
these  agents  the  largest  possible  descents  of  caloric. 

We  shall  conclude  by  showing  how  far  we  are  from  the  reali- 
zation, by  means  already  known,  of  all  the  motive  power  of  the 
combustibles. 

A  kilogram  of  coal  burned  in  the  calorimeter  furnishes  a 
quantity  of  heat  capable  of  raising  the  tempi-rat uru  of  about 
7000  kilograms  of  water  1  degree — that  is,  from  the  definition 
given  (page  43)  it  furnishes  7000  units  of  heat.  The  largest 
descent  of  caloric  which  can  be  realized  is  measured  by  the  dif- 
ference of  the  temperature  produced  by  combustion  and  that 
of  the  refrigerating  body.  It  is  difficult  to  see  any  limit  to  the 
temperature  of  combustion  other  than  that  at  which  the  com- 
bination of  the  combustible  with  oxygen  is  effected.  Let  us 
assume,  however,  that  this  limit  is  1000  degrees,  which  is  cer- 
tainly within  the  bounds  of  truth.  We  shall  assume  the  t<  m 
peratnre  of  the  refrigerator  to  be  0  degrees. 

\\  •  have  calculated  approximately  (page  4.r>)  the  quantity  of 
motive  power  developed  by  1000  units  of  In  at  in  passing  from 
the  temperature  100  to  the  temperature  99,  and  have  found 


THE  SECOND  LAW  OF  THERMODYNAMICS 

it  to  be  1.112  units,  each  equal  to  1  meter  of  water  raised  1 
meter. 

If  the  motive  power  were  proportional  to  the  descent  of  caloric, 
if  it  were  the  same  for  each  thermometric  degree,  nothing  would 
be  easier  than  to  estimate  it  from  1000  to  0  degrees.  Its  value 
would  be 

1.112  x  1000  =  1112. 

But  as  this  law  is  only  approximate,  and  perhaps  at  high 
temperatures  departs  a  good  deal  from  the  truth,  we  can  only 
make  a  very  rough  estimate.  Let  us  suppose  the  number  1112 
to  be  reduced  one-half — that  is,  to  560. 

Since  one  kilogram  of  coal  produces  7000  units  of  heat,  and 
since  the  number  560  is  referred  to  1000  units,  we  must  multi- 
ply it  by  7,  which  gives  us 

7  x  560  =  3920, 

which  is  the  motive  power  of  one  kilogram  of  coal. 

In  order  to  compare  this  theoretical  result  with  the  results 
of  experiment,  we  shall  inquire  how  much  motive  power  is  actu- 
ally developed  by  one  kilogram  of  coal  in  the  best  heat-engines 
known. 

The  engines  which  have  thus  far  offered  the  most  advanta- 
geous results  are  the  large  engines  with  two  cylinders  used  in 
the  pumping  out  of  the  tin  and  copper  mines  of  Cornwall.  The 
best  results  which  they  have  furnished  are  as  follows :  Sixty- 
five  million  pounds  of  water  have  been  raised  one  English  foot 
by  the  burning  of  one  bushel  of  coal  (the  weight  of  a  bushel  is 
88  Ibs.).  This  result  is  equivalent  to  raising  195  cubic  meters 
of  water  one  meter  by  the  use  of  one  kilogram  of  coal,  which 
consequently  produces  195  units  of  motive  power.* 

*  The  result  given  here  was  furnished  by  an  engine  whose  large  cylin- 
der was  35  inches  in  diameter,  with  a  7-foot  stroke  ;  it  is  used  to  pump 
out  one  of  the  mines  of  Cornwall,  called  "  Wheal  Abraham."  This  result 
should  in  a  way  be  considered  as  an  exception,  for  it  only  was  accomplished 
for  a  short  time  during  one  month.  A  product  of  30  million  Ibs.  raised 
one  English  foot  by  a  bushel  of  coal  weighing  88  Ibs.  is  generally  consid-( 
ered  to  be  an  excellent  result  for  a  steam-engine.  It  is  sometimes  reached 
by  the  engines  made  on  Watt's  system,  but  has  rnrely  been  exceeded. 
This  result  expressed  in  French  units  is  equal  to  104000  kilograms  raised 
one  meter  by  the  burning  of  one  kilogram  of  coal. 

By  what  we  ordinarily  understand  as  one  horse- power  in  the  calculation 


MEMOIRS    ON 

195  units  are  only  one-twentieth  of  3920,  the  theoretical  max- 
imum ;  consequently  only  ^  of  the  motive  power  of  the  combus- 
tible has  been  utilized. 

We  have,  moreover,  chosen  our  example  from  among  the 
best  steam-engines  known.  Most  of  the  others  have  been  vrry 
inferior.  For  example,  G'haillot's  engine  raises  '-.'<>  cubic  meters 
of  water  33  meters  in  consuming  30  kilograms  of  coal,  which  is 
equivalent  to  22  units  of  motive  power  to  1  kilogram,  a  result 
nine  times  less  than  that  cited  above,  and  one  hundred  and 
eighty  times  less  than  the  theoretical  maximum. 

We  should  not  expect  ever  to  employ  in  practice  all  the  mo- 
tive power  of  the  combustibles  used.  The  efforts  which  one 
would  make  to  attain  this  result  would  be  even  more  harmful 
than  useful  if  they  led  to  the  neglect  of  other  important  con- 
siderations. The  economy  of  fuel  is  only  one  of  the  conditions 
which  should  be  fulfilled  by  steam-engines  ;  in  many  cases  it  is 
only  a  secondary  consideration.  It  must  often  yield  the  prece- 
dence to  safety,  to  the  solidity  and  durability  of  the  engine,  to 
the  space  which  it  must  occupy,  to  the  cost  of  its  construction, 
etc.  To  be  able  to  appreciate  justly  in  each  case  the  consider- 
ations of  convenience  and  economy  which  present  themselves, 
to  be  able  to  recognize  the  most  important  of  those  which  are 
only  subordinate,  to  adjust  them  all  suitably,  and  finally  to 
reach  the  best  result  by  the  easiest  method — such  should  be  the 
power  of  the  man  who  is  called  on  to  direct  and  co-ordinate  the 
labors  of  his  fellow-men,  and  to  make  them  concur  in  attaining 
a  useful  purpose. 

BIOGRAPHICAL  SKETCH 

NICOLAS-LEON  A  RD-S  ADI  CARXOT  was  born  in  Paris  on  June 
1,  17%  ;  the  son  of  the  illustrious  engineer,  soldier,  and  states- 
man who  played  so  prominent  a  part  in  the  history  of  France 
during  the  Revolution.  He  was  educated  at  the  Keolc  I'i.l\ - 

of  the  efficiency  of  steam-engines,  a  10  horse-power  engine  should  raise 
10  x  75,  or  750  kilograms  1  meter  iu  a  second,  or  750  x  8600  =  3700000 
kilograms  1  meter  in  an  hour. 

If  we  suppose  eacli  kilogram  of  coal  to  raise  104000  kilograms,  it  is 
necessary  to  divide  2700000  by  104000  to  find  the  coal  burned  in  one  h«ur 
by  the  10  horse-power  engine,  which  gives  us  YoY  =  28  kilograms.  Hut 
it  is  very  rare  that  a  10  horse-power  engine  consumes  less  than  26  kilograms 
of  coal  an  hour. 

60 


THE    SECOND    LAW    OF    THERMODYNAMICS 

technique,  and  served  for  several  years  as  an  officer  of  engineers 
and  on  the  general  staff.  His  inclinations  towards  the  study  of 
science  were  so  strong  that  they  controlled  the  whole  course  of 
his  life.  While  still  engaged  in  his  profession  he  devoted  such 
time  as  he  could  spare  to  scientific  investigations,  and  he  at 
last  resigned  from  the  army  in  order  to  obtain  more  leisure  for 
studious  pursuits.  He  died  of  the  cholera  on  August  24,  1832. 
The  memoir  on  the  "Motive  Power  of  Heat"  is  the  only  one 
which  he  published.  It  shows  that  he  possessed  a  mind  able 
to  penetrate  to  the  heart  of  a  question,  and  to  invent  general 
methods  of  reasoning.  The  extracts  from  his  note-book,  pub- 
lished by  his  brother,  indicate  that  he  was  also  fertile  in  devis- 
ing experiments.  It  is  interesting  to  note  that  the  doubt  of 
the  validity  of  the  substantial  theory  of  heat,  expressed  by  him 
in  his  memoir,  developed  later  into  complete  disbelief,  and  that 
he  not  only  adopted  the  mechanical  theory  of  heat,  but  planned 
experiments  to  test  it  similar  to  those  of  Joule,  and  calculated 
that  the  mechanical  equivalent  of  heat  is  equal  to  370  kilogram- 
meters. 


ON  THE  MOTIVE  POWER  OF  HEAT,  AND 

ON   THE    LAWS   WHICH   CAN    BE 

DEDUCED  FROM  IT  FOR  THE 

THEORY  OF  HEAT 

BY 

R.  CLAUSIUS 


(Poggendorff's  Annalen,  vol.  Ixxix.,  pp.  376  and  500.     1850) 


CONTEXTS 

MM 

Work  of  Carnot  and  Clapeyron 65 

Dynamical  Theory  of  Heat 66 

Equivalence  of  Heat  and  Work 67 

Camot't  Cycle 

Application  to  Change  of  State 78 

Second  Late  of  Thermodynamics 88 

Carnot't  /•*«/<»•//»« 90 

Application  of  Clapeyron'*  Equation •.):{ 

Mechanical  Equivalent,  of  Heat 105 


ON  THE  MOTIVE  POWER  OF  HEAT,  AND 

ON  THE  LAWS  WHICH  CAN  BE 

DEDUCED  FROM  IT  FOR  THE 

THEORY  OF  HEAT 

BY 

R.   CLAUSIUS 

SINCE  heat  was  first  used  as  a  motive  power  in  the  steam- 
engine,  thereby  suggesting  from  practice  that  a  certain  quantity 
of  work  may  be  treated  as  equivalent  to  the  heat  needed  to  pro- 
duce it,  it  was  natural  to  assume  also  in  theory  a  definite  rela- 
tion between  a  quantity  of  heat  and  the  work  which  in  any 
possible  way  can  be  produced  by  it,  and  to  use  this  relation  in 
drawing  conclusions  about  the  nature  and  the  laws  of  heat  it- 
self. In  fact,  several  fruitful  investigations  of  this  sort  have 
already  been  made  ;  yet  I  think  that  the  subject  is  not  yet  ex- 
hausted, but  on  the  other  hand  deserves  the  earnest  attention 
of  phvsicists,  partly  because  serious  objections  can  be  raised  to 
the  conclusions  that  have  already  been  reached,  partly  because 
other  conclusions,  which  may  readily  be  drawn  and  which  will 
essentially  contribute  to  the  establishment  and  completion  of 
the  theory  of  heat,  still  remain  entirely  unnoticed  or  have  not 
yet  been  stated  with  sufficient  definiteness. 

The  most  important  of  the  researches  here  referred  to  was 
that  of  S.  Carnot,*  and  the  ideas  of  this  author,  were  after- 
wards given  analytical  form  in  a  very  skilful  way  by  Clapey- 
ron.f  Carnot  showed  that  whenever  work  is  done  by  heat  and 

*  Reflexions  sur  la  puissance  motrice  du  feu,  et  sur  Us  machines  propres  a 
developper  cette  puissance,  par  8.  Carnot.  Paris,  1824.  I  have  not  been 
able  to  obtain  a  copy  of  this  book,  and  am  acquainted  with  it  only  through 
the  work  of  Clapeyron  and  Thomson,  from  the  latter  of  whom  are  quoted 
the  extracts  afterwards  given. 

f  Journ.  de  I'ficole  Poly  technique,  vol.  xix.  (1834),  and  Pogg.  Ann.,  vol.  lix. 
E  65 


MKMoIKS    ON 

no  permanent  change  occurs  in  the  condition  of  the  work  in  <r 
body,  a  certain  quantity  of  heat  passes  from  a  hotter  to  a  c«>|.|rr 
body.  In  the  steam-engine,  for  example,  by  means  of  the 
steam  which  is  developed  in  the  boiU'r  and  precipitated  in  the 
condenser,  heat  is  transferred  from  the  grate  to  the  eonden>« T. 
This  transfer  he  considered  as  the  heat  change,  correspond  in;: 
to  the  work  done,  lie  says  expressly  that  no  heat  is  lost  in  tin- 
process,  but  that  the  quantity  of  heat  remains  unchanged,  and 
adds  :  "  This  fact  is  not  doubted  ;  it  was  assumed  at  first  with- 
out investigation,  and  then  established  in  many  cases  by  calori- 
metric  measurements.  To  deny  it  would  overthrow  tin-  whole 
theory  of  heat,  of  which  it  is  the  foundation."  I  am  nut  a  wan-. 
however,  that  it  has  been  sufficiently  proved  by  experiment 
that  no  loss  of  heat  occurs  when  work  is  done  ;  it  may,  perhaps. 
on  the  contrary,  be  asserted  with  more  correctness  that  even  if 
such  a  loss  has  not  been  proved  directly,  it  has  yet  been  sh»>wn 
by  other  facts  to  be  not  only  admissible,  but  even  highly  prob- 
able. If  it  be  assumed  that  heat,  like  a  substance,  cannot 
diminish  in  quantity,  it  must  also  be  assumed  that  it  cannot 
increase.  It  is,  however,  almost  impossible  to  explain  the  heat 
produced  by  friction  except  as  an  increase  in  the  quantity  of 
heat.  The  careful  investigations  of  Joule,  in  whieh  heat  is 
produced  in  several  different  ways  by  the  application  of  me- 
chanical work,  have  almost  certainly  proved  not  only  the  pos- 
sibility of  increasing  the  quantity  of  heat  in  any  ciivumst 
but  also  the  law  that  the  quantity  of  heat  developed  i-  propor- 
tional to  the  work  expended  in  the  operation.  To  this  it  mu.-t 
be  added  that  other  facts  have  lately  become  known  \\hich 
support  the  view,  that  heat  is  not  a  substance,  but  con>i.-i>  in  a 
motion  of  the  least  parts  of  bodies.  If  this  view  is  correct,  it 
is  admissible  to  apply  to  heat  the  general  mechanical  principle 
that  a  motion  may  be  transformed  into  work,  and  in  siu-h  a 
manner  that  the  loss  of  /•/.->•  rira  is  proportional  to  the  work  ac- 
complished. 

These  facts,  with  which  Carnot  also  was  well  acquainted,  and 
the  importance  of  which  he  has  expressly  re« -oirnixi-d.  almost 
compel  us  to  accept  the  equivalence  between  heat  and  work,  on 
the  modified  hypothesis  that  the  accomplishment  of  work  re- 
quires not  merely  a  change  in  the  >iistribution  of  heat,  hut  also 
an  actual  consumption  of  heat,  ami  that,  conversely,  heat  can 
be  developed  again  by  the  expenditure  of  work. 
'  M 


THE    SECOXD    LAW    OF    THERMODYNAMICS 

In  a  memoir  recently  published  by  Holtzmann,*  it  seems  at 
first  as  if  the  author  intended  to  consider  the  matter  from  this 
latter  point  of  view.  He  says  (p.  7)  :  "  The  action  of  the  heat 
supplied  to  the  gas  is  either  an  elevation  of  temperature,  in 
conjunction  with  an  increase  in  its  elasticity,  or  mechanical 
work,  or  a  combination  of  both,  and  the  mechanical  work  is 
the  equivalent  of  the  elevation  of  temperature.  The  heat  can 
only  be  measured  by  its  effects  ;  of  the  two  effects  mentioned 
the  mechanical  work  is  the  best  adapted  for  this  purpose,  and 
it  will  accordingly  be  so  used  in  what  follows.  I  call  the  unit 
of  heat  the  heat  which  by  its  entrance  into  a  gas  can  do  the  me- 
chanical work  a — that  is,  to  use  definite  units,  which  can  lift  a 
kilograms  through  1  meter."  Later  (p.  12)  he  also  calculates 
the  numerical  value  of  the  constant  a  in  the  same  way  as  Mayer 
had  already  done,f  and  obtains  a  number  which  corresponds 
with  the  heat  equivalent  obtained  by  Joule  in  other  entirely 
different  ways.  In  the  further  extension  of  his  theory,  how- 
ever, in  particular  in  the  development  of  the  equations  from 
which  his  conclusions  are  drawn,  he  proceeds  exactly  as  Clapey- 
ron  did,  so  that  in  this  part  of  his  work  he  tacitly  assumes  that 
the  quantity  of  heat  is  constant. 

The  difference  between  the  two  methods  of  treatment  has 
been  much  more  clearly  grasped  by  W.  Thomson,  who  has  ex- 
tended Carnot's  discussion  by  the  use  of  the  recent  observations 
of  Regnault  on  the  tension  and  latent  heat  of  water  vapor.  J  He 
speaks  of  the  obstacles  which  lie  in  the  way  of  the  unrestricted 
assumption  of  Carnot's  theory,  calling  special  attention  to  the 
researches  of  Joule,  and  also  raises  a  fundamental  objection 
which  may  be  made  against  it.  Though  it  may  be  true  in  any 
case  of  the  production  of  work,  when  the  working  body  has  re- 
turned to  the  same  condition  as  at  first,  that  heat  passes  from  a 
warmer  to  a  colder  body,  yet  on  the  other  hand  it  is  not  gener- 
ally true  that  whenever  heat  is  transferred  work  is  done.  Heat 
can  be  transferred  by  simple  conduction,  and  in  all  such  cases, 
if  the  mere  transfer  of  heat  were  the  true  equivalent  of  work, 
there  would  be  a  loss  of  working  power  in  Nature,  which  is 
hardly  conceivable.  Nevertheless,  he  concludes  that  in  the 

*  Ueber  die  Warme  und  Elasticit&t  der  Oase  und  Dampfe,  von  C.  Holtz- 
mann,  Mannheim,  1845  :  also  Pogg.  Ann.,  vol.  72a. 

f  Ann.  der  Chem.  und  Pfiarm.  of  WOhler  and  Liebig,  vol.  xlii.,  p.  239. 
J  Transactions  of  Uie  Royal  Society  of  Edinburgh,  vol.  xvi. 
67 


MEMOIRS    ON 

present  state  of  the  science  the  principle  adopted  by  Carnot  is 
still  to  he  taken  as  the  most  probable  basis  for  an  investigation 
of  the  motive  power  of  heat,  saying  :  "  If  we  abandon  this  prin- 
ciple, we  meet  with  innumerable  other  difficulties — insuperable 
without  further  experimental  investigation,  and  an  entire  it-- 
construction of  the  theory  of  heat  from  its  foundation."* 

I  believe  that  we  should  not  be  daunted  by  these  difficulties, 
but  rather  should  familiarize  ourselves  as  much  as  possible 
with  the  consequences  of  the  idea  that  heat  is  a  motion,  since 
it  is  only  in  this  way  that  we  can  obtain  the  means  \\here\\iih 
to  confirm  or  to  disprove  it.  Then,  too,  I  do  not  think  the 
difficulties  are  so  serious  as  Thomson  does,  since  even  though 
we  must  make  some  changes  in  the  usual  form  of  presentation, 
yet  I  can  find  no  contradiction  with  any  proved  facts.  It  is 
not  at  all  necessary  to  discard  Caruot's  theory  entirely,  a 
step  which  we  certainly  would  find  it  hard  to  take,  since  it 
has  to  some  extent  been  conspicuously  verified  by  experience. 
A  careful  examination  shows  that  the  new  method  does  not 
stand  in  contradiction  to  the  essential  principle  of  Carnot,  but 
only  to  the  subsidiary  statement  that  no  hmt  />•  l^t,  sim-e  in 
the  production  of  work  it  may  very  well  be  the  case  that  at  the 
same  time  a  certain  quantity  of  heut  is  consumeti  and  another 
quantity  transferred  from  a  hotter  to  a  colder  body,  and  hoth 
quantities  of  heat  stand  in  a  definite  relation  to  the  work  that 
is  done.  This  will  appear  more  plainly  in  the  sequel,  and  it 
will  there  be  shown  that  the  consequences  drawn  fmm  the  two 
assumptions  are  not  only  consistent  with  one  another,  but  are 
even  mutually  confirmatory. 

1.   CONSEQUENCES    OF    THE    PRINCIPLE    OP    THE    EQl'IY  \U.\-  K 
OF   HEAT   A  Xli    \\oKK 

We  shall  not  consider  here  the  kind  of  motion  which  ran  lie 
conceived  of  as  taking  place  within  bodies,  further  than  to  as- 
sume in  general  that  the  particle-;  of  l>odie>an-  in  motion,  ami 
that  their  heat  is  the  measure  of  their  ris  viva,  or  rath,  r  still 
more  generally,  we  shall  only  lay  down  a  principle  condition.-.! 
by  that  assumption  as  a  fundamental  prineiple,  in  the  words: 
In  all  cases  in  which  work  is  produced  by  the  agency  of  li.at. 
a  quantity  of  heat  is  consumed  which  is  proportional  to  the 

*  Math,  and  Phyt.  I\ij»r»,  vol.  I.,  p.  119,  note. 
68 


THE    SECOND    LAW    OF    THERMODYNAMICS 

work  done  ;  and,  conversely,  by  the  expenditure  of  an  equal 
quantity  of  work  an  equal  quantity  of  heat  is  produced. 

Before  we  proceed  to  the  mathematical  treatment  of  this 
principle,  some  immediate  consequences  may  be  premised 
which  affect  our  whole  method  of  treatment,  and  which  may 
be  understood  without  the  more  definite  demonstration  which 
will  be  given  them  later  by  our  calculations. 

It  is  common  to  speak  of  the  total  heat  of  bodies,  especially 
of  gases  and  vapors,  by  which  term  is  understood  the  sum  of 
the  free  and  latent  heat,  and  to  assume  that  this  is  a  quantity 
dependent  only  on  the  actual  condition  of  the  body  considered, 
so  that,  if  all  its  other  physical  properties,  its  temperature,  its 
density,  etc.,  are  known,  the  total  heat  contained  in  it  is  com- 
pletely determined.  This  assumption,  however,  is  no  longer 
admissible  if  our  principle  is  adopted.  Suppose  that  we  are 
given  a  body  in  a  definite  state — for  example,  a  quantity  of  gas 
with  the  temperature  t0  and  the  volume  t'0 — and  that  we  subject 
it  to  various  changes  of  temperature  and  volume,  which  are 
such,  however,  as  to  bring  it  at  last  to  its  original  state  again. 
According  to  the  common  assumption,  its  total  heat  will  again 
be  the  same  as  at  first,  from  which  it  follows  that  if  during 
one  part  of  its  changes  heat  is  communicated  to  it  from  with- 
out, the  same  quantity  of  heat  must  be  given  up  by  it  in  the 
other  part  of  its  changes.  Now  with  every  change  of  volume 
a  certain  amount  of  work  must  be  done  by  the  gas  or  upon  it, 
since  by  its  expansion  it  overcomes  an  external  pressure,  and 
since  its  compression  can  be  brought  about  only  by  an  exertion 
of  external  pressure.  If,  therefore,  among  the  changes  to  which 
it  has  been  subjected  there  are  changes  of  volume,  work  must 
be  done  upon  it  and  by  it.  It  is  not  necessary,  however,  that 
at  the  end  of  the  operation,  when  it  is  again  brought  to  its 
original  state,  the  work  done  by  it  shall  on  the  whole  equal 
that  done  upon  it,  so  that  the  two  quantities  of  work  shall 
counterbalance  each  other.  There  may  be  an  excess  of  one  or 
the  other  of  these  quantities  of  work,  since  the  compression 
may  take.place  at  a  higher  or  lower  temperature  than  the  ex- 
pansion, as  will  be  more  definitely  shown  later  on.  To  this 
excess  of  work  done  by  the  gas  or  upon  it  there  must  corre- 
spond, by  our  principle,  a  proportional  excess  of  heat  consumed 
or  produced,  and  the  gas  cannot  give  up  to  the  surrounding 
medium  the  same  amount  of  heat  as  it  receives. 


MKMnlRS    OX 

The  same  contradiction  to  the  ordinary  assumption  about 
the  total  hi-at  may  be  presented  in  another  way.  If  tin-  gas  at 
/„  and  i'0  is  brought  to  the  higher  temperature  /,  and  tin-  larger 
volume  t>j,  the  quantity  of  heat  which  must  be  imparted  to  it 
is,  on  that  assumption,  independent  of  the  way  in  which  the 
change  is  brought  about ;  from  our  principle,  however,  it  is  dif- 
ferent, according  as  the  gas  is  first  heated  wink-  its  vohun 
is  constant,  and  then  allowed  to  expand  at  the  <•< instant  tem- 
peratnre  /,,  or  is  first  expanded  at  the  constant  temperature/,,. 
and  then  heated,  or  as  the  expansion  and  heating  aiv  inter- 
changed in  any  other  way  or  even  occur  together,  since  in  all 
these  cases  the  work  done  by  the  gas  is  different. 

In  the  same  way,  if  a  quantity  of  water  at  the  temperature 
/„  is  changed  into  vapor  at  the  temperature  /,  and  of  the 
volume  r,.  it  will  make  a  difference  in  the  amount  of  lu-at 
needed  if  the  water  as  such  is  first  heated  to  /,  and  then 
evaporated,  or  if  it  is  evaporated  at  /„  and  the  vapor  then 
brought  to  the  required  volume  and  temperature.  /-,  and  /,.  or 
finally  if  the  evaporation  occurs  at  any  intermediate  tempera- 
ture. 

From  these  considerations  and  from  the  immediate  applica- 
tion of  the  principle,  it  may  easily  be  seen  what  conception 
must  he  formed  of  luti-nt  heat.  I'sing  again  the  example  al- 
ready employed,  we  distinguish  in  the  quantity  of  heat  which 
must  be  imparted  to  the  water  during  its  changes  the  t'rtr  and 
the  Inti-nt  heat.  Of  these,  however,  we  may  consider  only  the 
former  as  really  present  in  the  vapor  that  has  l.ecii  formed. 
The  latter  is  not  merely,  as  its  name  implies.  r«//, >/,/,,/  from 
our  perception,  hut  it  is  n<nr}n  /•<•  jm:«cnf  ;  it  is  nmsitinnl  during 
the  changes  in  doing  work. 

In  the  heat  consumed  we  must  still  introduce  :i  distinction  — 
that  is  to  say,  the  work  done  is  of  two  kinds,  l-'irst.  there  i>  a 
certain  amount  of  work  done  in  overcoming  the  mutual  attrac- 
tions of  the  particles  of  the  water,  and  in  M -parating  them  to 
such  a  distance  from  one  another. that  they  are  in  the  Mate  of 
vapor.  Secondly,  the  vapor  during  its  evolution  iyu>t  push 
back  an  external  pressure  in  order  to  make  room  for  itself. 
The  former  work  we  shall  call  the  i/i/rnml.  the  latter  the  «t- 
ln-H'il  work,  and  shall  partition  the  latent  heat  accordingly. 

It  can  make  no  difference  with  respect  to  the  //,/,  /  W  work 
whether  the  evaporation  goes  on  at  /0  or  at  /,,  or  at  any  intcr- 
70 


THE   SECOND   LAW   OF   THERMODYNAMICS 

mediate  temperature,  since  we  must  consider  the  attractive 
force  of  the  particles,  which  is  to  be  overcome,  as  invariable.* 

The  external  work,  on  the  other  hand,  is  regulated  by  the 
pressure  as  dependent  on  the  temperature.  Of  course  the 
same  is  true  in  general  as  in  this  special  example,  and  there- 
fore if  it  was  said  above  that  the  quantity  of  heat  which  must 
be  imparted  to  a  body,  to  bring  it  from  one  condition  to  an- 
other, depended  not  merely  on  its  initial  and  final  conditions, 
but  also  on  the  way  in  which  the  change  takes  place,  this 
statement  refers  only  to  that  part  of  the  latent  heat  which  cor- 
responds to  the  external  work.  The  other  part  of  the  latent 
heat,  as  also  the  free  heat,  are  independent  of  the  way  in  which 
the  changes  take  place. 

If  now  the  vapor  at  tl  and  v,  is  again  transformed  into  water, 
work  will  thereby  be  expended,  since  the  particles  again  yield 
to  their  attractions  and  approach  each  other,  and  the  external 
pressure  again  advances.  Corresponding  to  this,  heat  must  be 
produced,  and  the  so-called  liberated  heat  which  appears  during 
the  operation  does  not  merely  come  out  of  concealment  but  is 
actually  made  new.  The  heat  produced  in  this  reversed  opera- 
tion need  not  be  equal  to  that  used  in  the  direct  one,  but  that 
part  which  corresponds  to  the  external  work  may  be  greater  or 
less  according  to  circumstances. 

We  shall  now  turn  to  the  mathematical  discussion  of  the  sub- 
ject, in  which  we  shall  restrict  ourselves  to  the  consideration  of 
the  permanent  gases  and  of  vapors  at  their  maximum  density, 
since  these  cases,  in  consequence  of  the  extensive  knowledge 
we  have  of  them,  are  most  easily  submitted  to  calculation,  and 
besides  that  are  the  most  interesting. 

Let  there  be  given  a  certain  quantity,  say  a  unit  of  weight, 
of  a  permanent  gas.  To  determine  its  present  condition,  three 
magnitudes  must  be  known :  the  pressure  upon  it,  its  volume, 

'•  *  It  cannot  he  raised,  as  an  objection  to  this  statement,  that  water  at  t, 
has  less  cohesion  than  at.  tt,  and  that  therefore  less  work  would  be  needed 
to  overcome  it.  For  a  certain  amount  of  work  is  used  in  diminishing  the 
cohesion,  which  is  done  while  the  water  as  such  is  heated,  and  this  must 
be  reckoned  in  with  that  done  during  the  evaporation.  It  follows  at  once 
that  only  a  part  of  the  heat,  which  the  water  takes  up  from  without  whilfe 
it  is  being  heated,  is  to  be  considered  as  free  heat,  while  the  remainder  is 
used  in  diminishing  the  cohesion.  This  view  is  also  consistent  with  the 
circumstance  that  water  has  so  much  greater  a  specific  heat  than  ice,  and 
probably  also  than  its  vapor. 

71 


MEMOIRS   ON 

and  its  temperature.  These  magnitudes  are  in  a  mutual  re- 
lationship, which  is  expressed  by  the  combined  laws  of  Mariotte 
and  Gay-Lussac*,  and  may  be  represented  by  the  equation : 

(I.)  pv=R(a+t), 

where  p,  v,  and  t  represent  the  pressure,  volume,  and  tem- 
perature of  the  gas  in  its  present  condition,  a  is  a  constant. 
the  same  for  all  gases,  and  /,'  is  also  a  constant,  which  in  its 

complete  form  is  ~^i  if  j»o»  *><>»  ant*  'o  are  tne  corresponding 

values  of  the  three  magnitudes  already  mentioned  for  any  other 
condition  of  the  gas.  This  last  constant  is  in  so  far  different 
for  the  different  gases  that  it  is  inversely  proportional  to  their 
specific  gravities. 

It  is  true  that  Regnault  has  lately  shown,  by  a  very  careful 
investigation,  that  this  law  is  not  strictly  accurate,  yet  the  de- 
partures from  it  are  in  the  case  of  the  permanent  gases  \erv 
small,  and  only  become  of  consequence  in  the  case  of  those 
gases  which  can  be  condensed  into  liquids.  From  this  it  seems 
to  follow  that  the  law  holds  with  greater  accuracy  the  more 
removed  the  gas  is  from  its  condensation  point  with  respect  to 
pressure  and  temperature.  We  may  therefore,  while  the  ac- 
curacy of  the  law  for  the  permanent  gases  in  their  ordinary 
condition  is  so  great  that  it  can  be  treated  as  complete  in  most 
investigations,  think  of  a  limiting  condition  for  each  gas.  in 
which  the  accuracy  of  the  law  is  actually  complete.  We  shall. 
in  what  follows,  when  we  treat  the  permanent  gases  as  sin-h. 
assume  this  ideal  condition. 

According  to  the  concordant  investigations  of  Regnault  and 

Magnus,  the  value  of  -  for  atmospheric  air  is  equal  to  0.003G65, 

if  the  temperature  is  reckoned  in  centigrade  degrees  from  the 
freezing-point.  Since,  however,  as  has  been  mentioned,  the 
gases  do  not  follow  the  M.  and  G-.  law  exactly,  the  same  value 

of  -  will   not  always  be  obtained,  if  the  measurements  are 

made  in  different  circumstances.  The  number  here  given 
holds  for  the  case  when  air  is  taken  at  0°  under  the  pressure  of 
one  atmosphere,  and  heated  to  100°  at  constant  volume,  ami  the 

*  This  law  will  hereaflcr.  for  brevity,  be  called  tin-  M.  ami  <}   law,  anil 
Muriuiu-'a  law  will  be  called  the  M.  law. 
78 


THE   SECOND   LAW   OF   THERMODYNAMICS 

increase  of  its  expansive  force  observed.  If,  on  the  other  hand, 
the  pressure  is  kept  constant,  and  the  increase  of  its  volume 
observed,  the  somewhat  greater  number  0.003670  is  obtained. 
Further,  the  numbers  increase  if  the  experiment  is  tried 
under  a  pressure  higher  than  the  atmospheric  pressure,  while 
they  diminish  somewhat  for  lower  pressures.  It  is  not  there- 
fore possible  to  decide  with  certainty  on  the  number  which 
should  be  adopted  for  the  gas  in  the  ideal  condition  in  which 
naturally  all  differences  must  disappear ;  yet  the  number 
0.003665  will  surely  not  be  far  from  the  truth,  especially  since 
this  number  very  nearly  obtains  in  the  case  of  hydrogen,  which 
probably  approaches  the  most  nearly  of  all  the  gases  the  ideal 
condition,  and  for  which  the  changes  are  in  the  opposite  sense 
to  those  of  the  other  gases.  If  we  therefore  adopt  this  value 

of  —  we  obtain 

In  consequence  of  equation  (I.)  we  can  treat  any  one  of  the 
three  magnitudes  p,  v,  and  t — for  example,  p — as  a  function  of 
the  two  others,  v  and  t.  These  latter  then  are  the  independent 
variables  by  which  the  condition  of  the  gas  is  fixed.  We  shall 
now  seek  to  determine  how  the  magnitudes  which  relate  to  the 
quantities  of  heat  depend  on  these  two  variables. 

If  any  body  changes  its  volume,  mechanical  work  will  in  general 
be  either  produced  or  expended.  It  is,  however,  in  most  cases 
impossible  to  determine  this  exactly,  since  besides  the  external 
work  there  is  generally  an  unknown  amount  of  internal  work  done. 
To  avoid  this  difficulty,  Carnot  employed  the  ingenious  method 
already  referred  to  of  allowing  the  body  to  undergo  its  various 
changes  in  succession,  which  are  so  arranged  that  it  returns  at 
last  exactly  to  its  original  condition.  In  this  case,  if  internal 
work  is  done  in  some  of  the  changes,  it  is  exactly  compensated 
for  in  the  others,  and  we  may  be  sure  that  the  external  work, 
,  which  remains  over  after  the  changes  are  completed,  is  all  the 
work  that  has  been  done.  Clapeyron  has  represented  this  proc- 
ess graphically  in  a  very  clear  way,  arid  we  shall  follow  his  pres- 
entation now  for  the  permanent  gases,  with  a  slight  alteration 
rendered  necessary  by  our  principle. 

In  the  figure,  let  the  abscissa  oe  represent  the  volume  and 
the  ordinate  ea  the  pressure  on  a  unit  weight  of  gas,  in  a  con- 
dition in  which  its  temperature  =  t.  We  assume  that  the  gas 
73 


MEMOIRS    ON 


Fig.  I 


is  contained  in  an  extensible  envelope,  which,  however,  cannot 
exchange  heat  with  it.  If,  now.  it  is  allowed  to  expand  in  this 
envelope,  its  temperature  would  fall  if  no  heat  were  imparted 
to  it.  To  avoid  this,  let  it  be  put  in  contact,  during  its  ex- 
pansion, with  a  body,  A,  which  is  kept  at  the  constant  tempera- 
ture t,  and  which  imparts 
just  so  much  heat  to  the 
gas  that  its  temperature  also 
remains  equal  tot.  During 
this  expansion  at  constant 
temperature,  its  pressure 
diminishes  according  to  the 
M.  law,  and  may  be  repre- 
sented by  the  ordinate  of  a 
curve,  ab,  which  is  a  por- 
tion of  an  equilateral  hy- 
perbola. When  the  volume 
of  the  gas  has  increased  in  this  way  from  oe  to  of,  tin-  body  .1 
is  removed,  and  the  expansion  is  allowed  to  continue  with- 
out the  introduction  of  more  heat.  The  temperature  will 
then  fall,  and  the  pressure  diminish  more  rapidly  than  before. 
The  law  which  is  followed  in  this  part  of  the  process  may  In- 
represented  by  the  curve  be.  When  the  volume  of  the  gas  has 
increased  in  this  way  from  of  to  Of/,  and  its  temperature  has 
fallen  from  t  to  T,  we  begin  to  compress  it,  in  order  to  restore 
it  again  to  its  original  volume  oe.  If  it  were  left  to  itself  its 
temperature  would  airain  rise.  This,  however,  we  do  not  per- 
mit, but  bring  it  in  contact  with  a  body,  li,  at  the  constant  tem- 
perature r,  to  which  it  at  once  gives  up  the  heat  that  is  pro- 
duced, so  that  it  keeps  the  temperature  T  ;  and  while  it  is  in 
contact  with  this  body  we  compress  it  so  far  (by  the  amount  ////) 
that  the  remaining  compression  he  is  exactly  sufficient  to  raise 
its  temperature  from  T  to  /.  if  during  this  last  coinpn -<-ion  it 
gives  up  no  heat.  During  the  former  compression  the  piv.-smv 
increases  according  to  the  M.  law.  and  is  represented  l,\  tic 
portion  cd  of  an  equilateral  hyperbola.  During  the  latter,  on 
the  other  hand,  the  increase  is  more  rapid  and  is  represented 
by  the  curve  da.  This  curve  must  end  exactly  at  n.  for  since 
at  the  end  of  the  operation  the  volume  and  temperature  have 
again  their  original  values,  the  same  must  be  true  of  the 
pressure  also,  which  is  a  function  of  them  both.  The  gas  is 
74 


THE    SECOND    LAW    OF    THERMODYNAMICS 


therefore  in  the  same  condition  again  as  it  was  at  the  begin- 
ning. 

Now,  to  determine  the  work  produced  by  these  changes,  for 
the  reasons  already  given,  we  need  to  direct  our  attention  only 
to  the  external  work.  During  the  expansion  the  gas  does  work, 
which  is  determined  by  the  integral  of  the  product  of  the  dif- 
ferential of  the  volume  into  the  corresponding  pressure,  and 
is  therefore  represented  geometrically  by  the  quadrilaterals 
eabf  and  fbcg.  During  the  compression,  on  the  other  hand, 
work  is  expended,  which  is  represented  similarly  by  the  quad- 
rilaterals (jcdk  and  hdae.  The  excess  of  the  former  quantity  of 
work  over  the  latter  is  to  be  looked  on  as  the  whole  work  pro- 
duced during  the  changes,  and  this  is  represented  by  the  quad- 
rilateral abed. 

If  the  process  above  described  is  carried  out  in  the  reverse 
order,  the  same  magnitude,  abed,  is  obtained  as  the  excess  of 
the  work  expended  over  the  work  done. 

In  order  to  make  an  analytical  application  of  the  method 
just  described,  we  will  assume  that  all  the  changes  which  the 
gas  undergoes  are  infinitely  small.  We  may  then  treat  the 
curves  obtained  as  straight  lines,  as  they  are  represented  in  the 
accompanying  figure.  \Ve  may  also,  in  determining  the  area 
of  the  quadrilateral 
abed,  consider  it  a  par- 
allelogram, since  the 
error  arising  there- 
from can  only  be  a 
quantity  of  the  third 
order,  while  the  area 
itself  is  a  quantity  of 
the  second  order.  On 
this  assumption,  as 
may  easily  be  seen, 
the  area  may  be  repre- 
sented by  the  product 
ef.bk,  if  k  is  the  point  in  which  the  ordinate  bf  cuts  the  lower 
side  of  the  quadrilateral.  The  magnitude  bk  is  the  increase 
of  the  pressure,  while  the  gas  at  the  constant  volume  of  has 
its  temperature  raised  from  r  to  t — that  is,  by  the  differential 
t  —  T  =  dt.  This  magnitude  may  be  at  once  expressed  by  the 
aid  of  equation  (I.)  in  terms  of  v  and  t,  and  is 
75 


Fig.  2 


MEMO  IKS    ON 
W 

*—  T 

If,  further,  we  denote  the  increase  of  volume  ef  by  dv,  we  ob- 
tain the  area  of  the  quadrilateral,  and  so,  also, 


(1)  The  work  done  = 

We  must  now  determine  the  heat  consumed  in  these  changes. 
The  quantity  of  heat  which  must  be  communicated  to  a  gas, 
while  it  is  brought  from  any  former  condition  in  a  definite  way 
to  that  condition  in  which  its  volume  =  v  and  its  temperature 
=  t,  may  be  called  Q,  and  the  changes  of  volume  in  the  above 
process,  which  must  here  be  considered  separately,  may  l>e  rep- 
resented as  follows:  efby  dv,  fig  by  d'v,  eh  by  St>,  and  fgby  3'r. 
During  an  expansion  from  the  volume  oe  —  v  to  the  volume 
of  —  v  4-  dv  at  the  constant  temperature  /,  the  gas  must  receive 
the  quantity  of  heat 


di- 

and  correspondingly,  during  an  expansion  from  oh  —  v  +  2t»  to 
og  =  v  +  2r  +  d'v  at  the  temperature  t  —  dl,  the  quantity  of 
heat, 

[8^®-K»-l* 


In  the  case  before  us  this  latter  quantity  must  be  taken  as 
negative  in  the  calculation,  because  the  real  process  was  a  com- 
pression instead  of  the  expansion  assumed.  Pnrini:  tin-  expan- 
sion from  of  to  og  and  the  compression  from  oh  to  oe,  the  gas 
h:is  neither  gained  nor  lost  heat,  and  hence  the  quantity  of 
heat  which  the  gas  has  received  in  excess  of  that  which  it  has 
given  up  —  that  is,  the  heat  m,  ,*„,„,,/ 


The  magnitudes  Iv  and  d'v  must  be  eliminate'!  fn"n  this  ex- 
pression. For  this  purpose  we  have  first,  immediately  from  the 
inspection  of  the  figure,  the  following  equation  : 


l-'r<>in  the  condition  that  during  the  compression  from  «//  to 
ml  therefore  also  conversely  during  an  expansion  from  <><• 
to  oh  occurring  under  the  same  conditions,  and  similarly  dur- 
76 


THE    SECOND    LAW    OF    THERMODYNAMICS 

ing  the  expansion  from  of  to  og,  both  of  which  occasion  a  fall 
of  temperature  by  the  amount  dt,  the  gas  neither  receives  nor 
gives  up  heat,  we  obtain  the  equations 


Eliminating  from  these  three  equations  and  equation  (2)  the 
three  magnitudes  d'v,  cv,  and  I'v,  and  also  neglecting  in  the 
development  those  terms  which,  in  respect  of  the  differentials, 
are  of  a  higher  order  than  the  second,  we  obtain 

(3)       The  heat  consumed  =    -L-  ((-¥  1  —  4-  ( (-rr  I   dvdt. 
L  d t  \  dv  }      dv  \  a  t  }  J 

If  we  now  return  to  our  principle,  that  to  produce  a  certain 
amount  of  work  the  expenditure  of  a  proportional  quantity  of 
heat  is  necessary,  we  can  establish  the  formula 
The  heat  consumed  _   . 

The  work  done 

where  A  is  a  constant,  whicli  denotes  the  heat  equivalent  for 
the  unit  of  work.  The  expressions  (1)  and  (3)  substituted  in 
this  equation  give 


R.dvdt 

v 
or 


(IL)  dt  \dv)      dv  \dt 

We  may  consider  this  equation  as  the  analytical  expression 
of  our  fundamental  principle  applied  to  the  case  of  permanent 
gases.  It  shows  that  Q  cannot  be  a  function  of  v  and  t,  if 
these  variables  are  independent  of  each  other.  For  if  it  were, 
then  by  the  well-known  law  of  the  differential  calculus,  that  if 
a  function  of  two  variables  is  differentiated  with  respect  to  botji 
of  them,  the  order  of  differentiation  is  indifferent,  the  right- 
hand  side  of  the  equation  should  be  equal  to  zero. 

The  equation  may  also  be  brought  into  the  form  of  a  complete 
differential  equation, 

77 


MK  MM  IKS     ON 


in  which  r  is  an  arbitrary  function  of  v  and  /.  This  differen- 
tial equation  is  naturally  not  integrable,  but  becomes  so  only 
if  a  second  relation  is  given  between  the  varial>K-s.  by  which  / 
may  be  treated  as  a  function  of  t'.  The  reason  for  this  is 
found  in  the  last  term,  and  this  corresponds  exactly  to  the 
>.rlirnal  work  done  during  the  change,  since  the  dilTerential  of 
this  work  is  ]><h\  from  which  we  obtain 


if  we  eliminate/)  by  means  of  (I.). 

We  have  thus  obtained  from  equation  (Il.rt)  what  was  in- 
troduced before  as  an  immediate  consequence  of  onr  principle. 
that  the  total  amount  of  heat  received  by  the  gas  durin.ir  a 
change  of  volume  and  temperature  can  be  separated  into  two 
parts,  one  of  which,  T,  which  comprises  the  //•>,•  heat  that  has 
entered  and  the  heat  consumed  in  doing  internal  work,  if  any 
such  work  has  been  done,  has  the  properties  which  are  com- 
monly assigned  to  the  total  heat,  of  being  a  function  of  /•  and  /. 
and  of  being  therefore  fully  determined  by  the  initial  and  final 
conditions  of  the  gas,  between  which  the  transformation  has 
taken  place;  while  the  other  part,  which  comprises  the  heat 
consumed  in  doing  external  work,  in  dependent  not  only  on  the 
terminal  conditions,  but  on  the  whole  course  of  the  changes 
between  these  conditions. 

Before  we  undertake  to  prepare  this  equation  for  further 
conclusions,  we  shall  develop  the  analytical  e\|iiv-.-ion  of  our 
fundamental  principle  for  the  case  of  vapors  at  their  maximum 
density. 

In  this  case  we  have  no  right  to  apply  the  M.  and  (i.  law,  and 
so  must  restrict  ourselves  to  the  principle  alone.  In  order  to 
obtain  an  equation  from  it,  we  again  use  the  method  given  by 
Carnot  and  graphically  presented  by  Clapcyion.  \\ith  a  slight 
modification.  Consider  a  liquid  contained  in  a  \c-el  impen- 
etrable by  heat,  of  which,  however,  only  a  part  is  filled  by 
the  liqni  I.  while  the  rest  is  left  free  for  the  vapor,  \\hirh  is  at 
the  maximum  density  corresponding  to  its  temperature./.  Tho 
total  volume  of  both  liquid  and  vapor  is  represented  in  the  ac- 
companying figure  by  the  abscissa  oe,  and  the  pressure  of  the 
78 


THE    SECOND    LAW    OF    THERMODYNAMICS 


vapor  by  the  ordinate  ea.  Let  the  vessel  now  yield  to  the 
pressure  and  enlarge  in  volume  while  the  liquid  and  vapor  are 
in  contact  with  a  body,  A, 
at  the  constant  temperature 
/.  As  the  volume  increases, 
more  liquid  evaporates,  but 
the  heat  which  thus  becomes 
latent  is  supplied  from  the 
body  A,  so  that  the  temper- 
ature, and  so  also  the  press- 
ure, of  the  vapor  remain 
unchanged.  If  in  this  way 


the  total  volume  is  increased 
from  oe  to  of,  an  amount  of  Fiff- 3 

external  work  is  done  which 

is  represented  by  the  rectangle  eabf.  Now  remove  the  body 
A  and  let  the  vessel  increase  in  volume  still  further,  while  heat 
can  neither  enter  nor  leave  it.  In  this  process  the  vapor  already 
present  will  expand,  and  also  new  vapor  will  be  produced,  and 
in  consequence  the  temperature  will  fall  and  the  pressure  dimin- 
ish. Let  this  process  go  on  until  the  temperature  has  changed 
from  t  to  r,  and  the  volume  has  become  oy.  If  the  fall  of  press- 
ure during  this  expansion  is  represented  by  the  curve  be,  the 
external  work  done  in  the  process  =fbcg. 

Now  diminish  the  volume  of  the  vessel,  in  order  to  bring  the 
liquid  with  its  vapor  back  to  its  original  total  volume,  oe;  and 
let  this  compression  take  place,  in  part,  in  contact  with  the  body 
B  at  the  temperature  r,  into  which  body  all  the  heat  set  free 
by  the  condensation  of  the  vapor  will  pass,  so  that  the  temper- 
ature remains  constant  and  =  r,  in  part  without  this  body,  so 
that  the  temperature  rises.  Let  the  operation  be  so  managed 
that  the  first  part  of  the  compression  is  carried  out  only  so  far 
(to  oh)  that  the  volume  he  still  remaining  is  exactly  such  that 
compression  through  it  will  raise  the  temperature  from  r  to  t 
again.  During  the  former  diminution  of  volume  the  pressure 
remains  invariable,  =  gc,  and  the  external  work  employed  is 
equal  to  the  rectangle  gcdh.  During  the  latter  diminution  of 
volume  the  pressure  increases  and  is  represented  by  the  curve 
da,  which  must  end  exactly  at  the  point  «,  since  the  original 
pressure,  ea,  must  correspond  to  the  original  temperature,  t. 
The  work  employed  in  this  last  operation  is  =  hdae.  At  the 

79 


MEMOIRS    ON 

end  of  the  operation  the  liquid  and  vapor  are  again  in  the  same 
condition  as  at  the  beginning,  so  that  the  excess  of  tin-  fj-tt-nml 
work  done  over  that  employed  is  also  the  total  work  done.  It 
is  represented  by  the  quadrilateral  abed,  and  its  area  must  also 
be  set  equal  to  the  heat  consumed  during  the  same  time. 

For  our  purposes  we  again  assume  that  the  changes  just  de- 
scribed are  infinitely  small,  and  on  this  assumption  represent 
the  whole  process  by  the  accompanying  figure,  in  which  the 
curves  ad  and  be  which  occur  in 
Fig.  3  have  become  straight  lines. 
So  far  as  the  area  of  the  quadrilat- 
eral abed  is  concerned,  it  may  again 
be  considered  a  parallelogram,  and 
may  be  represented  by  the  product 
ef.bk.  If,  now,  the  pressure  of  tin; 
vapor  at  the  temperature  t  and  at 


/    a        '    its  maximum  tension  is  represented 
Pig  4  by/?,  and  if  the  temperature  ditTcr- 

ence  t  —  T  is  represented  by  */-. 

hftve  »=$*. 

The  line  ef  represents  the  increase  of  volume,  which  occurs  in 
consequence  of  the  passage  of  a  certain  quantity  of  liquid, 
which  may  be  called  dm,  over  into  vapor.  Representing  now 
the  volume  of  a  unit  weight  of  the  vapor  at  its  maximum  <!en- 
sity  at  the  temperature  t  by  *,  and  the  volume  of  the  same 
quantity  of  liquid  at  the  temperature  /  by  9,  we  have  evidently 

ef-(s-0)dm, 
and  consequently  the  area  of  the  quadrilateral,  or 

(5)  The  work  done  =  (*  -  ef^dmdt. 

In  order  to  represent  the  quantities  of  heat  concerned,  we 
will  introduce  the  following  symbols.  The  quantity  of  heat 
which  becomes  latent  when  a  unit  weight  of  the  liquid  evapo- 
rates at  the  temperature  /  and  under  the  corresponding  press- 
ure, is  called  r,  and  the  specific  heat  of  the  liquid  is  called 

Both  of  these  quantities,  as  well  as  also  *,  a,  and  ^,  are  to  be 

considered  functions  of  t.     Finally,  let  us  denote  by  //»//  tin- 
quantity  of  heat  which  must  be  imparted  to  a  unit  weight  of 
80 


THE   SECOND   LAW    OF   THERMODYNAMICS 

the  vapor  if  its  temperature  is  raised  from  t  to  t  -f-  dt,  while  it  is 
so  compressed  that  it  is  again  at  the  maximum  density  for  this 
temperature  without  the  precipitation  of  any  part  of  it.  The 
quantity  7*  is  likewise  a  function  of  t.  It  will,  for  the  pres- 
ent, be  left  undetermined  whether  it  has  a  positive  or  negative 
value. 

If  we  now  denote  by  /i  the  mass  of  liquid  originally  present 
in  the  vessel,  and  by  tit  the  mass  of  vapor,  and  further  by  dm 
the  mass  which  evaporates  during  the  expansion  from  oe  to  of, 
and  by  d'm  the  mass  which  condenses  during  the  compression 
from  og  to  oh,  the  heat  which  becomes  latent  in  the  first  opera- 
tion and  is  taken  from  the  body  A  is 

rdm, 

and  that  which  is  set  free  in  the  second  operation  and  is  given 
to  the  body  B  is  j 


In  the  other  expansion  and  in  the  other  compression  heat  is 
neither  gained  nor  lost,  so  that,  at  the  end  of  the  process, 

(6)  The  heat  consumed=rdm—(r  —  jrdt}d'm. 

In  this  expression  the  differential  d'm  must  be  replaced  by  dm 
and  dt.  For  this  purpose  we  make  use  of  the  conditions  under 
which  the  second  expansion  and  the  second  compression  oc- 
curred. The  mass  of  vapor,  which  condenses  during  the  com- 
pression from  oh  to  oe,  and  which  would  be  evolved  by  the  cor- 
responding expansion  from  oe  to  oh,  may  be  represented  by  Sm, 
and  that  which  is  evolved  by  the  expansion  from  of  to  og  by 
I'm.  We  then  have  at  once,  since  at  the  end  of  the  process  the 
same  mass  of  liquid  p  and  the  same  mass  of  vapor  m  must  be 
present  as  at  the  beginning,  the  equation 

dm  +  Km  —  d'm  -f  Im. 

Further,  we  obtain  for  the  expansion  from  oe  to  oh,  since  in 
it  the  temperature  of  the  mass  of  liquid  p  and  the  mass  of  va- 
por m  must  be  lowered  by  dt  without  the  emission  of  heat,  the 
equation 


and  similarly  for  the  expansion  from  of  to  og,  by  substituting 
fi—dm  and  m+  dm  for  /*  and  m,  and  Z'm  for  §m, 


81 


M  KMOIRS    ON 

If  from  these  three  equations  and  (6)  we  eliminate  the  magni- 
tudes d'm,  3m,  and  I'm,  and  reject  terms  of  higher  order  than 
the  second,  we  have 


(7) 

The  formulas  (?)  and  (5)  must  now  he  connected  in  the  same 
way  as  that  used  in  the  case  of  the  permanent  gases,  that  is, 


and  we  obtain  as  the  analytical  expression  of  the  fundamental 
principle  in  the  case  of  vapors  at  their  maximum  density  the 
equation 

(III.)  £+e-h  =  A(*-.)'%. 

If,  instead  of  using  our  principle,  we  adopt  the  assumption 
that  the  quantity  of  heat  is  conx/anf.  \ve  must  replace  (III.),  as 
appears  from  (7),  by 

(8) 


This  equation  has  been  used,  if  not  exactly  in  the  same  form, 
at  least  in  its  general  sense,  to  obtain  a  value  for  tin-  magni- 
tude A.  So  long  as  Watt's  law  is  considered  true  for  water. 
that  the  sum  of  the  free  and  latent  heats  of  a  quantity  of  vapor 
at  its  maximum  density  is  equal  for  all  temperatures,  and  that 
therefore  ^r 

rf/+'=°; 

it  must  be  concluded  that  for  this  liquid  7<=0.  This  conclu- 
sion has,  in  fact,  often  been  stated  us  con-ret,  in  that  it  has 
been  said  that  if  a  quantity  of  vapor  is  at  its  maximum  dmsit  v. 
and  then  compressed  or  expanded  in  a  ves>i-l  impermeable  hy 
heat,  it  remains  at  its  maximum  .im.-itv.  \\\\\  sinee  Ki-nault* 
has  corrected  Watt's  law  by  substituting  for  it  the  approximate 
relation  ,/.. 


the  equation  (8)  gives  for  7*  the  value  0.305.     It  would  there- 
fore follow  that  the  quantity  of  vapor  formerly  considered  in 

•  mm.  de  rAcad..  t.  xxl.,  the  9tb  and  10.1,  M,  moires. 


THE    SECOND    LAW    OF    THERMODYNAMICS 

the  vessel  impermeable  by  heat  would  be  partly  condensed  by 
compression,  and  on  expansion  would  not  remain  at  the  maxi- 
mum density,  since  its  temperature  would  not  fall  in  a  way  to 
correspond  to  the  diminution  of  pressure. 

It  is  entirely  different  if  we  replace  equation  (8)  by  (III.). 
The  expression  on  the  right-hand  side  is,  from  its  nature,  always 
positive,  and  it  therefore  follows  that  h  must  be  less  than  0.305. 
It  will  subsequently  appear  that  the  value  of  this  expression  is 
so  great  that  h  is  negative.  We  must  therefore  conclude  that 
the  quantity  of  vapor  before  mentioned  is  partly  condensed, 
not  by  compression,  but  by  expansion,  and  that  by  compression 
its  temperature  rises  at  a  greater  rate  than  the  density  increases, 
so  that  it  does  not  remain  at  its  maximum  density. 

It  must  be  admitted  that  this  result  is  exactly  opposed  to 
the  common  view  already  referred  to ;  yet  I  do  not  believe  that 
it  is  contradicted  by  any  experimental  fact.  Indeed,  it  is  more 
consistent  than  the  former  view  with  the  behavior  of  steam 
as  observed  by  Pambour.  Pambour*  found  that  the  steam 
which  issues  from  a  locomotive  after  it  has  done  its  work 
always  has  the  temperature  at  which  the  tension,  observed 
at  the  same  time,  is  a  maximum.  From  this  it  follows 
either  that  A=0,  as  it  was  once  thought  to  be,  because  this 
assumption  agreed  with  Watt's  law,  accepted  as  probably  true, 
or  that  h  is  negative.  For  if  h  were  positive,  the  temperature 
of  the  vapor,  when  released,  would  be  too  high  in  comparison 
with  its  tension,  and  that  could  not  have  escaped  Pambour's 
notice.  If,  on  the  other  hand,  h  is  negative,  according  to  our 
former  statement,  there  can  never  arise  from  this  cause  too 
low  a  temperature,  but  a  part  of  the  steam  must  become  liquid, 
so  as  to  maintain  the  rest  at  the  proper  temperature.  This 
part  need  not  be  great,  since  a  small  quantity  of  vapor  sets  free 
on  condensation  a  relatively  large  quantity  of  heat,  and  the 
water  formed  will  probably  be  carried  on  mechanically  by  the 
rest  of  the  steam,  and  will  in  such  researches  pass  unnoticed, 
the  more  likely  as  it  might  be  thought,  if  it  were  to  be  observed, 
that  it  was  water  from  the  boiler  carried  out  mechanically. 

The  results  thus  far  obtained  have  been  deduced  from  the 
fundamental  principle  without  any  further  hypothesis.  The 
equation  (II. a)  obtained  for  permanent  gases  may,  however,  be 

*  Traite  des  Locomotives,  secoud  edition,  and  Theorie  des  Machines  d  Vupeur, 
second  edition. 


MEMOIRS    ON 

made  much  more  fruitful  by  the  help  of  an  obvious  subsidiary 
hypothesis.  The  gases  show  in  their  various  relation 
pecially  in  the  relation  expressed  by  the  M.  and  (J.  law  be- 
tween volume,  pressure,  and  temperature,  so  great  a  regularity 
of  behavior  that  we  are  naturally  led  to  take  the  view  that  the 
mutual  attraction  of  the  particles,  which  acts  within  solid  ami 
liquid  bodies,  no  longer  acts  in  gases,  so  that  while  in  the  case 
of  other  bodies  the  heat  which  produces  expansion  must  over- 
come not  only  the  external  pressure  but  the  internal  attraction 
as  well,  in  the  case  of  gases  it  has  to  do  only  with  the  external 
pressure.  If  this  is  the  case,  then  during  the  expansion  of  a 
gas  only  so  much  heat  becomes  latent  as  is  used  in  doing  ex- 
ternal work.  There  is,  further,  no  reason  to  think  that  a  gas. 
if  it  expands  at  constant  temperature,  contains  more  ir<-<-  heat 
than  before.  If  this  be  admitted,  we  have  the  law  :  a  per- 
manent gas,  when  expanded  at  constant  frt/t/irrti/nrr,  tnkcs  /</> 
only  xo  much  heat  ns  /.<  mnsumed  in  tl<n'n</  r.rfrrntil  imrk  ilnrimi 
the  expansion.  This  law  is  probably  true  for  any  <:as  with  the 
same  degree  of  exactness  as  that  attained  by  the  M.  and  (J.  law 
applied  to  it. 

From  this  it  follows  at  once  that 


since,  as  already  noticed,  R  -  dv  represents  the  external  work 
v 

done  during  the  expansion  dv.  It  follows  that  the  fimetion  (' 
which  occurs  in  (II.  a)  does  not  contain  /•.  and  the  equation 
therefore  takes  the  form 


(ll.b)  dQ=cdt+AR         <i'\ 

where  c  can  be  a  function  of  /  only.  It  is  even  probable  that 
this  magnitude  r,  which  represents  the  speciii.-  heat  of  the  gag 
at  constant  volume,  is  a  constant. 

Now  in  order  to  apply  this  equation  to  special  cases,  we  must 
introduce  the  relation  between  the  variable-  (J.  f,  and  /-.  which 
is  obtained  from  the  conditions  of  each  separate  ,-i-e.  mi«>  the 
equation,  and  so  make  it  integrable.  We  shall  here  <  ••.n.-ider 
only  a  few  simple  examples  of  this  sort,  which  are  eiiher  in- 
teresting in  themselves  or  become  so  by  comparison  with  other 
theorems  already  announced. 

84 


THE    SECOND    LAW    OF    THERMODYNAMICS 

We  may  first  obtain  the  specific  heats  of  the  gas  at  constant 
volume  and  at  constant  pressure  if  in  (II.  b)  we  set  #=const., 
and  j3  =  const.  In  the  former  case,  dv=Q,  and  (II.  b)  becomes 


In  the  latter  case,  we  obtain  from  the  condition  j9=const.,  by 
the  help  of  equation  (I.), 

,      Rdt 

*>=—  , 

or 

dv        tit 


v 
and  this,  substituted  in  (II.  b),  gives 


if  we  denote  by  c'  the  specific  heat  at  constant  pressure. 

It  appears,  therefore,  that  the  difference  of  the  two  specific 
heats  of  any  gas  is  a  constant  magnitude,  AR.  This  magni- 
tude also  involves  a  simple  relation  among  the  different  gases. 

The  complete  expression  for  R  is  -  —  )  ,  where  p0,  v0,  and  t0 


are  any  three  corresponding  values  of  p,  v,  and  t  for  a  unit 
of  weight  of  the  gas  considered,  and  it  therefore  follows,  as 
has  already  been  mentioned  in  connection  with  the  adoption 
of  equation  (I.),  that  R  is  inversely  proportional  to  the  specific 
gravity  of  the  gas,  and  hence  also  that  the  same  statement  must 
hold  for  the  difference  c'  —  c=AR,  since  A  is  the  same  for  all 
gases. 

If  we  reckon  the  specific  heat  of  the  gas,  not  with  respect  to 
the  unit  of  weight,  but,  as  is  more  convenient,  with  respect  to 
the  unit  of  volume,  we  need  only  divide  c  and  c'  by  v0,  if  the 
volumes  are  taken  at  the  temperature  t0  and  pressure  j»0.  Des- 
ignating these  quotients  by  •/  and  y,  we  obtain 

(11)  y' 


I'o 

In  this  last  quantity  nothing  appears  which  is  dependent  on 
the  particular  nature  of  the  gas,  and  the  difference  of  the  specific 
heats  referred  to  the  unit  of  volume  is  therefore  the  same  for  all 
gases. 

This  law  was  deduced  by  Clapeyron  from  Carnot's  theory, 
85 


MEMOIRS    ON 

though  the  constancy  of  the  difference  c'  —  c,  which  we  have 
deduced  before,  is  not  found  in  his  work,  where  the  expression 
given  for  it  still  has  the  form  of  a  function  of  the  temperature. 
If  we  divide  equation  (11)  on  both  sides  by  y,  we  have 

(12)  *-l=-.-J^-, 

in  which  k,  for  the  sake  of  brevity,  is  used  for  the  quotient  — , 

or,  what  amounts  to  the  same  thing,  for  the  quotient—.      This 

quantity  has  acquired  special  importance  in  science  from  the 
theoretical  discussion  by  Laplace  of  the  propagation  of  sound 
in  air.  The  excess  of  fin's  quotient  over  unity  is  therefore.  /<//•  t In- 
different yases,  inversely  /n-ojtortiontil  to  tin-  sped  lie  Ite/i/s  of  tin' 
same  at  constant  volume,  ifike$tttn  r>f<  mil  lo  tin  unit  ,,f  ruhnne. 
This  law  has,  in  fact,  been  found  by  Dulong  from  experiment* 
to  be  so  nearly  accurate  that  he  has  assumed  it,  in  view  of  its 
theoretical  probability,  to  be  strictly  accurate,  and  has  there- 
fore employed  it,  conversely,  to  calculate  the  specific  heats  of 
the  different  gases  from  the  values  of  k  determined  by  obser- 
vation. It  must,  however,  be  remarked  that  the  law  is  only 
theoretically  justified  when  the  M.  and  G.  law  holds,  whii-h  is 
not  the  case  with  sufficient  exactness  for  all  the  gases  employed 
by  Dulong. 

If  it  is  now  assumed  that  the  specific  heat  of  gases  at  con- 
stant volume  f  is  constant,  which  has  been  stated  al>ove  to  bo 
very  probable,  the  same  follows  for  the  specific  heat  at  con- 
stant pressure,  and  consequently  the  quotient  of  the  tir 


heats  C——k  is  a  constant.     This  law,  which  Poisson  has  already 
c 

assumed  as  correct  on  the  strength  of  the  experiment!  of  (lay- 
Lussac  and  Welter, and  has  made  the  basis  of  his  investigations 
on  the  tension  and  heat  of  gases,  f  is  therefore  in  ^<«><\  agree- 
ment with  our  present  theory,  while,  it  would  not  bo  possible 
on  Carnot's  theory  as  hitherto  developed. 

If  in  equation  (II. A)  we  set  @=const.,  we  obtain  the  follow- 
ing equation  between  v  and  / : 

•  Ann.  df  Chim.  et  de  Phyt.,  xli..  and  Po^g.  Ann.,  xvi. 
f  Traiti  de  Mecanique,  second  edition,  vol.  ii.,  p.  646. 
M 


THE   SECOND   LAW   OF   THERMODYNAMICS 

(13) 
which  gives,  if  c  is  considered  constaiit, 

v  —  '—.(a+t)=const., 

AH      c' 
or,  since  from  equation  (10«),^—  =  —  —  1  =  #—  1, 

C  C 

v*~l  (rt-M)=const. 

Hence  we  have,  if  v0,  t0,  and  j»0  are  three  corresponding  values 
of  M,  and/,,  a+j      /,o 

(14)  a  +  t»-\v 

If  we  substitute  in  this  relation  the  pressure  p  first  for  v  and 
then  for  t  by  means  of  equation  (I.),  we  obtain 


These  are  the  relations  which  hold  between  volume,  temper- 
ature, and  pressure,  if  a  quantity  of  gas  is  compressed  or  ex- 
panded within  an  envelope  impermeable  by  heat.  These  equa- 
tions agree  precisely  with  those  which  have  been  developed  by 
Poisson  for  the  same  case,*  which  depends  upon  the  fact  that 
he  also  treated  k  as  a  constant. 

Finally,  if  we  set  t  =  const,  in  equation  (Il.b),  the  first  term 
on  the  right  drops  out,  and  there  remains 

(17)  dQ=ARf^dv, 
from  which  we  have 

Q  =  AR  (a  +  t)  log  v  +  const., 

or,  if  we  denote  by  v0,  p0,  t0,  and  Q0  the  values  of  v,  p,  t,  and 
Q,  which  hold  at  the  beginning  of  the  change  of  volume, 

(18)  Q-Q0=AR(a  +  t0)\og?-. 

From  this  follows  the  law  also  developed  by  Carnot  :  If  a  gas 
changes  its  volume  without  changing  Us  temperature,  the  quanti- 
ties of  heat  evolved  or  absorbed  are  in  arithmetical  progression, 
while  the  volumes  are  in  geometrical  progression. 

*  Traite  de  Mecanique,  vol.  ii.,  p.  647. 
87 


M  KM  <•!  KS    OX 

Further,  if  we  substitute  for  R  in  (18)  the  complete  expres- 
sion "  ,  we  have 

(19)  Q-Q0=APor0\og^. 

If  now  we  apply  this  equation  to  the  different  gases,  not  by 
using  equal  weights  of  them,  but  such  quantities  as  have  at  tln> 
outset  equal  volumes,  t'0,  it  becomes  in  all  its  parts  indt-pen- 
tlent  of  the  special  nature  of  the  gas,  and  agrees  with  the 
well-known  law  which  Dulong  proposed,  guided  by  the  ahuve- 
nientioned  simple  relation  of  tin-  magnitude  k  —  1,  that  nil 
gases,  if  equal  volumes  of  tin  in  art'  taken  at  tin'  sunn'  /»•////</•/•///// n 
and  under  flu-  same  fires*  n  re,  ami  if  they  art'  then  finii/n-'-ssnf  <>r 
expanded  by  an  equal  fraction  <>f  their  minims,  fill- 
absorb  an  equal  quantify  <>f  heat.  Kipiation  (19)  is,  however, 
much  more  general.  It  states  in  addition,  that  the  quantity  of 
heat  is  independent  of  tin-  temperature  at  irhieh  flu-  ml  nine  oftkt 
yas  is  altered,  if  only  the  quantity  of  tin-  gas  employed  is  always 
determined  so  that  the  original  volume  r0  is  always  the  same 
at  the  different  temperatures  ;  and  it  states  further,  that  if  fhf 
original  pressure  is  ilijt'in-nf  in  the  <////'»/•<///  <•<»»>•.  ////  quantities 
<>f  heat  are  proportional  to  it. 

II.    CONSEQUENCES    OP    CARNOT's    PRIN'lIM.i:     IX     rnxxr.i  TH»x 
WITH   THE   ONE   ALREADY    IXTISiMH  «  I  l> 

Carnot  assumed,  as   has  already  ln-i-n    nifiitiuncd,  that   tin- 
equivalent  of  the  work  done  by  In-at  is  fnnn<l  in  tin'  mn-f  fr» 
of  heat  from  a  hotter  to  a  colder  body,  while  tin'  quantity  »;/ '  Itmt 
remains  undiminished. 

The  latter  part  of  this  assumption — namely,  that  the  quan- 
tity of  heat  remains  undimiiiislii'd— contradicts  our  former  prin- 
ciple, and  must  therefore  be  rejected  if  we  are  to  retain  that 
principle.  On  the  other  hand,  the  first  part  may  still  obtain  in 
all  its  essentials.  For  though  we  do  not  need  a  special  equiva- 
lent for  the  work  done,  since  we  have  assumed  as  such  an  act  nal 
consumption  of  heat,  it  still  may  well  be  possible  that  such  a 
transfer  of  heat  occurs  at  tin-  sunn-  /////»  as  the  consumption  of 
heat,  and  als<.  stands  in  a  definite  relation  to  the  work  done. 
It  becomes  important,  therefore,  to  consider  whether  this  as- 
sumption, besides  the  mere  possibility,  has  also  a  sufficient 
probability  in  its  favor. 


THE   SECOND    LAW    OF   THERMODYNAMICS 

A  transfer  of  heat  from  a  hotter  to  a  colder  body  always  oc- 
curs in  those  cases  in  which  work  is  done  by  heat,  and  in  which 
also  the  condition  is  fulfilled  that  the  working  substance  is  in 
the  same  state  at  the  end  as  at  the  beginning  of  the  operation. 
For  example,  we  have  seen,  in  the  processes  described  above 
and  represented  in  Figs.  1  and  3,  that  the  gas  and  the  evapo- 
rating water  took  up  heat  from  the  body  A  as  their  volume  in- 
creased, and  gave  it  up  to  the  body  B  as  their  volume  dimin- 
ished :  so  that  a  certain  quantity  of  heat  was  transferred  from 
A  to  B,  and  this  was  in  fact  much  greater  than  that  which  we 
assumed  to  be  consumed,  so  that  in  the  infinitely  small  changes, 
which  are  represented  in  Figs.  2  and  4,  the  latter  was  an  in- 
finitesimal of  the  second  order,  while  the  former  was  one  of 
the  first  order.  Yet,  in  order  to  establish  a  relation  between 
the  heat  transferred  and  the  work  done,  a  certain  restric- 
tion is  necessary.  For  since  a  transfer  of  heat  can  take  place 
without  mechanical  effect  if  a  hotter  and  a  colder  body  are  im- 
mediately in  contact  and  heat  passes  from  one  to  the  other  by 
conduction,  the  way  in  which  the  transfer  of  a  certain  quantity 
of  heat  between  two  bodies  at  the  temperatures  t  and  r  can  be 
made  to  do  the  maximum  of  work  is  to  so  carry  out  the  proc- 
ess, as  was  done  in  the  above  cases,  that  two  bodies  of  different 
temperatures  never  come  in  contact. 

It  is  this  maximum  of  work  which  must  be  compared  with 
the  heat  transferred.  When  this  is  done  it  appears  that  there 
is  in  fact  ground  for  asserting,  with  Carnot,  that  it  depends 
only  on  the  quantity  of  the  heat  transferred  and  on  the  tempera- 
tures t  and  T  of  the  two  bodies  A  and  B,  but  not  on  the  nature 
of  the  substance  by  means  of  which  the  work  is  done.  This 
maximum  has,  namely,  the  property  that  by  expending  it  as 
great  a  quantity  of  heat  can  be  transferred  from  the  cold  body 
B  to  the  hot  body  A  as  passes  from  A  to  B  when  it  is  produced. 
This  may  easily  be  seen,  if  we  think  of  the  whole  process  for- 
merly described  as  carried  out  in  the  reverse  order,  so  that,  for 
example,  in  the  first  case  the  gas  first  expands  by  itself,  until 
its  temperature  falls  from  t  tor,  is  then  expanded  in  connection 
with  B,  is  then  compressed  by  itself  until  its  temperature  is 
again  t,  and  finally  is  compressed  in  connection  with  A.  In  this 
case  more  work  will  be  employed  during  the  compression  than 
is  produced  during  the  expansion,  so  that  on  the  whole  there 
is  a  loss  of  work,  which  is  exactly  as  great  as  the  gain  of  work  in 


MEMOIRS    ON 

the  former  process.  Further,  there  will  be  just  as  much  heat 
taken  from  the  body  B  as  was  before  given  to  it,  and  just  a> 
much  given  to  the  body  A  as  was  before  taken  from  it,  whence  it 
follows  not  only  that  the  same  amount  of  heat  is  produced  as  was 
formerly  consumed,  but  also  that  the  heat  which  in  the  former 
process  was  transferred  from  A  to  B  now  passes  from  B  to  A. 

If  we  now  suppose  that  there  are  two  substances  of  which  the 
one  can  produce  more  work  than  the  other  by  the  transfer  of  a 
given  amount  of  heat,  or,  what  comes  to  the  same  thing,  needs 
to  transfer  less  heat  from  A  to  B  to  produce  a  given  <|ii;mtity 
of  work,  we  may  use  these  two  substances  alternately  by  pro- 
ducing work  with  one  of  them  in  the  above  process,  and  by  e\- 
pending  work  upon  the  other  in  the  reverse  process.  At  the 
end  of  the  operations  both  bodies  are  in  their  original  condi- 
tion ;  further,  the  work  produced  will  haVe  exactly  count  er- 
balanced  the  work  done,  and  therefore,  by  our  former  principle, 
the  quantity  of  heat  can  have  neither  increased  nor  diminished. 
The  only  change  will  occur  in  the  ttixtrilmtion  of  the  heat,  since 
more  heat  will  be  transferred  from  B  to  A  than  from  A  to  B, 
and  so  on  the  whole  heat  will  be  transferred  from  B  to  A.  By 
repeating  these  two  processes  alternately  it  would  be  possible, 
without  any  expenditure  of  force  or  any  other  change,  to  trans- 
fer as  much  heat  as  we  please  from  a  cold  to  a  hot  body,  and  this 
is  not  in  accord  with  the  other  relations  of  heat,  since  it  always 
shows  a  tendency  to  equalize  temperature  differences  and 
therefore  to  pass  from  hotter  to  colder  bodies. 

It  seems,  therefore,  to  be  theoretically  admissible  to  retain 
the  first  and  the  really  essential  part  of  Carnot's  assumptions, 
and  to  apply  it  as  a  second  principle  in  conjunction  with  the 
first ;  and  the  correctness  of  this  method  is,  as  we  shall  soon 
see,  established  already  in  many  cases  by  its  consequences.  ' 

On  this  assumption  we  may  express  the  maximum  of  work 
which  can  be  produced  by  the  transfer  of  a  unit  of  heat  from  t  lie 
body  A  at  the  temperature  t  to  the  body  B  at  the  temperature 
r,  as  a  function  of  /  and  r.  The  value  of  this  function  must 
naturally  be  smaller  as  the  difference  t—r  is  smaller,  and  when 
this  is  infinitely  small  (=dt)  it  must  go  over  into  the  pni.l- 
uct  of  dt  and  a  function  of  t  only.  For  this  latter  case,  with 
which  we  will  concern  ourselves  for  the  present,  the  work  may 

be  expressed  by  the  form  (,-d(,  where  C  is  a  function  of  /  only. 
90 


THE  SECOND  LAW  OF  THERMODYNAMICS 

In  order  to  apply  this  result  to  the  permanent  gases,  we  re- 
turn to  the  process  represented  in  Fig.  2.  In  that  case  the 
quantity  of  heat, 


passed  during  the  first  expansion  from  A  to  the  gas,  and  by  the 
first  compression  the  part  of  it  expressed  by 


or  by 

(dQ\  [d/(lQ\  (I  S<lQ\-]7  1f 
[  i  —  )«'"  —  ~rA  ~i  —  )  —  T  —  (  ~r~  )  uvclt, 
\dv  )  \-dt\dv  )  dv  \  dt  )  J 

was  given  up  to  the  body  B.  The  latter  magnitude  is,  there- 
fore, the  quantity  of  heat  transferred.  Since  we  may  neglect 
the  term  of  the  second  order  with  respect  to  the  one  of  the 
first  order,  we  retain  simply 


The  work  produced  at  the  same  time  was 
Rdv.dt 

v 

and  we  can  thus  at  once  form  the  equation 
Rdv.dt 


'^*> 


or, 


If,  in  the  second  place,  we  make  a  similar  application  to  the 
process  represented  in  Fig.  4  relating  to  vaporization,  we  have 
for  the  quantity  of  heat  carried  from  A  to  B 


or  rdm  —  i^-4-c  —  h  \dmdt, 

\dt  I 

for  which,  by  neglecting  the  term  of  the  second  order,  we  may 
set  simply 

rdm. 
91 


.MEMOIRS    ON 
The  work  produced  was 


and  we  therefore  get  the  equation 


or, 

(v.)  rs-a(,_.)& 

These  are  the  two  analytical  expressions  of  Carnot's  principle, 
as  they  are  given  by  Clapeyron  in  his  memoir,  in  a  somewhat 
different  form.  For  vapors  he  stops  with  this  equation  (V.) 
and  some  immediate  applications  of  it.  For  gases,  on  the  other 
hand,  he  makes  the  equation  (IV.)  the  basis  of  a  more  extended 
development.  It  is  in  this  development  that  the  partial  dis- 
agreement appears  between  his  results  and  ours. 

We  shall  now  connect  these  two  equations  with  the  results  of 
the  first  principle,  first  considering  equation  (IV.)  in  connec- 
tion with  the  consequences  formerly  deduced  for  the  case  of 
permanent  gases. 

If  we  restrict  ourselves  to  that  result  which  depends  only  on 
the  fundamental  principle  —  that  is,  to  equation  (II.  a)  —  we  can 
use  equation  (IV.)  to  further  define  the  magnitude  /',  which 
appears  there  as  an  arbitrary  function  of  -v  ami  /,  and  our  equa- 
tion becomes 


(II.c)       d 

where  B  is  now  an  arbitrary  function  of  /  only. 

If  we  also  accept  as  correct  the  subsidiary  hypothetic,  then 
equation  (IV.)  is  not  necessary  for  the  further  definition  of 
(Il.a)  ;  since  the  same  end  is  more  completely  attained  by 
e.  |  nation  (9),  which  followed  as  an  immediate  consequence  of 
this  hypothesis  in  connection  with  the  first  principle.  \\  ,• 
gain,  however,  an  opportunity  to  subject  the  results  of  the  two 
principles  to  a  comparative  test.  Equation  (9)  reads  : 
dQ\  R.A(a+t) 


and  if  wo  compare  this  with  (IV.),  we  see  that  they  both  ex- 

press the  same  result,  only  the  one  in  a  more  definite  way  than 

92 


THE    SECOND    LAW    OF    THERMODYNAMICS 

the  other,  since  for  the  general  temperature  function  denoted 
in  (IV.)  by  C,  the  equation.  (9)  gives  the  special  expression 
A  (a  +  t). 

To  this  striking  agreement  it  may  be  added  that  equation 
(V.),  in  which  also  the  function  C  appears,  confirms  the  view 
that  A  (a  +  t)  is  the  correct  expression  for  this  function.  This 
equation  has  been  used  by  Clapeyron  and  Thomson  to  calculate 
the  values  of  C  for  several  temperatures.  Clapeyron  chose  as 
the  temperatures  the  boiling-points  of  ether,  alcohol,  water, 
and  oil  of  turpentine,  arid  by  substituting  inequation  (V.)  the 

values  of  -~,  s,  and  r  for  these  liquids,  determined  by  experi- 
ments at  these  boiling-points,  he  obtained  for  C  the  numbers 
contained  in  the  second  column  of  the  table  which  follows. 
Thomson,  on  the  other  hand,  considered  water  vapor  only,  but 
at  different  temperatures,  and  thence  calculated  the  value  of  C 
for  every  degree  between  0°  and  230°  Cent.  For  this  purpose 
Kegnault's  series  of  observations  have  given  him  an  admissible 

basis  so  far  as  the  magnitudes  -j-  and  r  are  concerned  ;  but  the 

magnitude  s  is  not  so  well  known  for  other  temperatures  as  for 
the  boiling-point,  and  about  this  magnitude  Thomson  felt  him- 
self compelled  to  make  an  assumption,  which  he  himself  rec- 
ognized as  only  approximately  correct,  and  considered  as  a 
temporary  aid,  to  be  employed  until  more  exact  data  are  de- 
termined— namely,  that  water  vapor  at  its  maximum  density 
follows  the  M.  and  G.  law.  The  numbers  which  follow  from 
his  calculation  for  the  same  temperatures  as  those  used  by 
Clapeyron  are  given  in  the  third  column  reduced  to  French 
units  : 

I 


1 

t  IN  CENT.  DEGRKE8 

2 

C  ACCORDING  TO 
CLAPEYKON 

3 

C  ACCORDING   TO 
THOMSON 

35°.5 

78°.8 
100° 
156°.S 

0.733 

0.828 
0.897 
0.930 

0.728 
0.814 
0.855 
0.952 

It  appears  that  the  values  of  0  found  in  both  cases  in- 
crease slowly  with  the  temperature,  similarly  to  the  values 


IfKMOIBS    OH 

of  A  (a+t).  They  are  in  the  ratio  of  tlic  numbers  in  the  fol- 
lowing rows  :  i:i.i:>:i.-  :!.••; 

1  :  1.12  :  1.17  :  1.31 

and  if  we  determine  the  ratios  of  the  values  of  A  (a  +  /)  corre- 
sponding to  the  same  temperatures,  we  obtain 

1  :  1.14  :  1.21  :  1.39. 

This  series  of  relative  values  diverges  from  the  two  others  only 
so  far  as  can  be  accounted  for  by  the  uncertainty  of  the  data 
which  underlie  them.  The  same  agreement  will  be  shown 
later  in  connection  with  the  determination  of  the  constant  A, 
in  respect  to  the  absolute  values. 

Such  an  agreement  between  results  which  are  obtained  from 
entirely  different  principles  cannot  be  accidental  ;  it  rather 
serves  as  a  powerful  confirmation  of  the  two  principles  and  the 
first  subsidiary  hypothesis  annexed  to  them. 

Returning  again  to  the  application  of  equations  (IV.)  and  (  V.  i. 
we  may  remark  that  the  former,  so  far  as  relates  to  the  per- 
manent gases,  has  only  served  to  confirm  conclusions  already 
obtained.  In  the  consideration  of  vapors,  and  of  all  other  sub- 
stances to  which  Carnot's  principle  will  In-  applied  in  the  future. 
it  furnishes,  however,  an  essential  improvement,  in  that  it  per- 
mits us  to  replace  the  function  C,  which  recurs  everywhere,  by 
the  definite  expression  A  («+/). 

Uy  this  substitution  equation  (V.)  becomes 


and  we  therefore  obtain  for  a  vapor  a  simple  relation  between 
the  temperature  at  which  it  is  formed,  the  pressure,  the  vol- 
ume, and  the  latent  heat.  This  we  can  use  in  drawing  further 
conclusions. 

If  the  M.  and  G.  law  were  accurate  for  vapors  at  their  maxi- 
mum density,  wo  should  have 

<••»,,  /»  =  /?(<!  +  /). 

Kliminating  the  magnitude  *  from  (\  .a)  by  the  use  of  this 
(•((nation,  and  neglecting  the  magnitude  «r.  which  vanishes  in 
'  -mparidon  with  *  if  the  temperature  is  not  very  high,  we  ob- 


THE    SECOND    LAW    OF    THERMODYNAMICS 

If  we  make  the  further  assumption  that  r  is  constant,  we  ob- 
tain by  integration,  if  pl  denotes  the  tension  of  the  vapor  at 

100°'  ,„&  __  r«-100) 

*>  pl    ^.#(a-HlOO)(rt-M) 


or  if  we  set/  -100  =  r,  «  +  100  =  a,  and  -.- 

A. 


This  equation  cannot,  of  course,  be  accurate,  since  the  two 
assumptions  made  in  its  development  are  not  accurate  ;  but 
since  these,  at  least  to  a  certain  extent,  approach  the  truth,  the 

quantity  -~-  will  roughly  represent  the  value  of  the  quantity 

log  —  .     .We  may  explain  in  this  way  how  it  happens  that  this 

relation,  if  the  constants  a  and  /3,  instead  of  having  values 
given  them  depending  on  their  definitions,  are  considered  as 
arbitrary,  may  serve  as  an  empirical  formula  for  the  calculation 
of  vapor  tensions,  without  our  being  compelled  to  consider  it 
as  fully  proved  by  theory,  as  is  sometimes  done. 

The  most  immediate  application  of  equation  (V.«)  is  to 
water  vapor,  for  which  we  have  the  largest  collection  of  experi- 
mental data,  in  order  to  investigate  how  far  it  departs,  when  at 
its  maximum  density,  from  the  M.  and  G.  law.  The  magnitude 
of  this  departure  cannot  be  unimportant,  since  carbonic  acid 
and  sulphurous  acid,  even  at  temperatures  and  tensions  at 
which  they  are  still  far  removed  from  their  condensation  points, 
show  noticeable  departures. 

Equation  (V.)  may  be  put  in  the  following  form  : 


The  expression  here  found  on  the  left-hand  side  would  be 
very  nearly  constant,  if  the  M.  and  G.  law  were  applicable, 
since  this  law  would  give  immediately,  from  (20), 


and  s  —  a  can  be  substituted  for  s  in  this  equation  with  approxi- 

mate accuracy.     This  expression  is,  therefore,  especially  suited 

95 


M  KM  "I  us    ON 

to  show  clearly  any  departure  from  the  M.  and  G.  law,  from 
tlie  examination  of  its  true  values  as  they  may  be  calculated 
from  the  expression  on  the  right-hand  side  of  (22).      I   ha\e 
carried  out  this  calculation  for  a  series  of  temperatures, 
fur  /•  and  p  the  numbers  given  by  Regnault.* 

First  with  respect  to  the  Intrnt  heat:  Regnanlt  states f  that 
the  quantity  of  heat  X.  which  must  be  imparted  to  a  unit  of 
weight  of  water,  in  order  to  heat  it  from  0°  to  t°  and  then  \<> 
evaporate  it  at  that  temperature,  may  be  represented  with 
tolerable  accuracy  by  the  formula  : 

(23)  X  =  GOG.  5 +0.305  t. 

But  now,  from  the  significance  of  X, 


(23«)  \  =  r+  I   < 


and  for  the  magnitude  c,  the  specific  heat  of  water,  which  ap- 
pears in  this  formula,  Regnault  has  given  the  formula  :  J 


By  the  help  of  these  two  equations  we  obtain  for  the  latent, 
heat  from  equal  ion  (23)  the  expression  : 

(24)      r  =  G06.5—  0.  695.  *-0.  00002.  /'  —  0.0000003./'.  5? 

Second,  with  respect  to  the  pressure  :  in  order  to  obtain 
from  his  numerous  observations  the  most  probable  values. 
lleirnaultjl  made  use  of  a  graphic  representation,  by  construct- 
ing curves,  of  which  the  abscissas  represented  the  temperature 
and  the  ordinates  the  pressure  />.  and  which  arc  drawn  in  sec- 
tions from  —33°  to  +230°.  From  100°  to  230°  he  has  al.-o 

*  Mem.  de  VAcad,  de  I'Intt.  de  France,  vol.  xxi.  (1847). 

f  Ibid..  Mem.  ix.;  also  Fogg  Ann..  lid.  98.  f  Ibid..  M'm.  x. 

£  In  most  of  his  investigations  Ri-gnaiill  lias  not  so  much  C>I»<TY<-<I  un- 
bent which  becomes  latent  by  evaporation  of  the  vapor  us  that  which  he 
COOtCT  free  by  its  condensation,  ami,  therefore,  since  it  lias  ln-m  slmwii 
above  thai,  if  tbe  principle  of  the  equivalence  of  heat  and  work  is  mtr«  t, 
the  quantity  of  beat  which  a  quantity  of  vapor  gives  up  on  coixlciis.-itinu 
iii-i-il  not  ulvvays  In-  the  same  as  tbat  wbicli  it  almorbs  during  ii>  fnrina- 
tion.  tbe  question  may  arise,  whether  such  differences  may  not  ba\<  in 
tered  in  Ib-cnniilt's  experiments,  so  that  tin-  formula  given  for  r  would 
IM-COIHC  in  idinidsiblc.  I  b«-li(-vc  that  we  may  answer  this  question  in  tin- 
negative,  since  Itegnault  BO  arranged  his  experiments  that  tin-  «>n.|.nvi. 
tion  of  the  vapor  occurred  under  the  same  pressure  as  its  formation—  that 
U.  nearly  under  the  pressure  which  corresponded  as  a  maximum  to  th<- 
observed  teni|>erntiire.  and  in  tb  is  case  Just  us  much  ln-at  IIIUM  hi-  •  volvi-,1 
by  condensation  as  is  absorbed  by  evaporation.  |  Ibid.,  .!/•<//.  viii. 

M 


THE    SECOND    LAW    OF    THERMODYNAMICS 


drawn  a  curve,  of  which  the  ordinates  represent  not  p  itself, 
but  the  logarithms  of  p.  From  this  presentation  the  following 
values  have  been  taken,  which  are  to  be  considered  as  the  im- 
mediate results  of  his  observations,  while  the  other  more  com- 
plete tables  contained  in  the  memoir  were  calculated  from 
formulas,  of  which  the  choice  and  determination  depended  in 
the  first  instance  upon  these  values : 

II 


t  IN  DEGREES 

t  IN  DEGREES 

CENTIGKADE 

p  IN 

CENTIGRADE 

ON  THE  AIR- 
THERMOMETER 

METERS 

ON  THE  AIR- 
THERMOMETER 

—20° 

0.91 

110° 

—  10 

2.08 

120 

0 

4.60 

130 

10 

9.16 

140 

20 

17.39 

150 

30 

31.55 

160 

40 

54.91 

170 

50 

91.98 

180 

60 

148.79 

190 

70 

233.09 

200 

80 

354.64 

210 

90 

525.45 

220 

100 

760.00 

230 

p  IN  MILLIMETERS 

FROM  THE 

FROM  THE 

CURVE  OP 

CURVE  OP 

NUMBERS 

LOGARITHMS* 

1073.7 

1073.3 

1489.0 

1490.7 

2029.0 

2030.5 

2713.0 

2711.5 

3572,0 

3578.5 

4647.0 

4651.6 

5960.0 

5956.7 

7545.0 

7537.0 

9428.0 

9425.4 

11660.0 

11679.0 

14308.0 

14325.0 

17390.0 

17390.0 

20915.0 

20927.0 

Now  in  order  to  carry  out  with  these  data  the  calculation  in 

hand,  I  first  determined  from  these  tables  the  values  of r. 

p  dt 

for  the  temperatures  —15°,  —  5P,  5°,  15°,  etc.,  in  the  following 
way.  Since  the  magnitude  —~  only  diminishes  slowly  as  the 

temperature  rises,  I  have  considered  as  uniform  the  diminution 
in  each  interval  of  10°,  say  from  -20°  to  -10°,  from  —10°  to 
0°,  etc.,  so  that  I  could  look  on  the  value  holding,  for  example, 
for  25°  as  the  mean  of  all  the  values  holding  between  20°  and 


30°.     On  this  assumption,  since—  "ji— 
the  formula : 


d  (log  p) 
dt 


could  use 


*  Instead  of  the  logarithms  obtained  immediately  from  the  curve  and 
adopted  by  Regnault,  the  numbers  corresponding  to  them  are  given,  in 
order  to  facilitate  comparison  with  the  numbers  in  the  next  coluiuu. 
G  97 


MEMOIRS    ON 


_  log  p*r  -log  Ptf 
10 


—  Log  Pao" 
10.  Jf 

in  which  Log  indicates  the  Briggsiau  logarithms  and  ^f  the 

modulus  of  this  system.     By  help  of  these  values  of  -•-.'  and 

the  values  of  r  given  by  equation  (24),  and  of  the  valu< 
for  a,  the  values  which  the  expression  on  the  right-hand  side 

of  (22),  and  so  also  the  expression  Ap  (s  —  a}  -——  ,  take  for  tilt- 

temperatures  —  15°,  —5°,  5°,  etc.,  were  calculated  and  aiv 
given  in  the  accompanying  table.  For  temperatures  above  100° 
both  series  of  numbers  given  for  p  are  used  separatt-ly.  and  the 
two  results  found  in  each  case  given  opposite  each  other.  The 
significance  of  the  third  and  fourth  columns  will  be  indicated 
in  the  sequel. 


Ill 


t  IX   DKHRKKM 
OOTWUM 

OS   THK    AIK 


FROM  TH1  OBttRTKD 
VALCW 


nu>M  «QC  ATIOX  (37) 


Dirriarccn 


-15 
-5 
5 
15 
25 
85 
45 
.v, 
65 
75 
OB 
95 
105 
115 
125 
100 
145 
155 
165 
175 
185 
195 

m 

215 
225 


00.S1 
8098 
00.00 

::n  in 


:t.Mii 
00.40 


•J'.t  -s 


OO.SO 

30^00 

•J'.t  ss 
•J'.t  7«J 


•j'..  .;.-, 


29.47    2950 
00.10    00.00 


•j*  ss      -J'.l  1,1 

•J*  III 


•J'.l  1  7 
•js  '.t't 

•J*    M, 


08.01  •J-'  i'.» 

27.84  27.90 

27.76  27.67 

03  r.  WIM 


•j.;  :,•;    -jr,  711 
•j.j  :,u 


28.14 

•J7  -'.« 

•J7  t;-j 

•J7  :u 
•J7  n-j 
•j.;  .is 


0.00 

+  1  :::: 
-11  17 

-0.10 

-0.08 

0.00 

+  002 

0.00 

0.00 

-0.02 

-  0.01 

-0.14-0.17 

+  0.01  +015 

+  0.10  +  006 

-0.08      "  Jl 

-  0.05  +  0.20 

+  0.18 

• 

+  0.05  -  0.01 
-0.14-0.05 
-0.12+0.18 
+  0.18  +  0.08 
+  0.12-011 
-0.82-0.18 


H 


THE    SECOND    LAW    OF    THERMODYNAMICS 

It  appears  at  once  from  this  table  that  Ap  (s—o)  —  ^-7-  is  not 

constant  as  it  should  be  if  the  M.  and  Gr.  law  were  applicable, 
but  diminishes  distinctly  as  the  temperature  rises.  Between 
35°  and  90°  this  diminution  appears  to  be  very  uniform.  Under 
35°,  especially  in  the  region  of  0°,  there  appear  noticeable 
irregularities,  which,  however,  may  be  simply  explained  from 
the  fact  that  in  that  region  the  pressure  p  and  its  differential 

coefficient  ~  are  very  small,  and  therefore  small  errors,  which 

(it 

fall  quite  within  the  limits  of  the  errors  of  observation,  may 
become  relatively  important.  It  may  be  added  that  the  curve 
by  which  the  separate  values  of  p  are  determined,  as  mentioned 
above,  is  not  drawn  in  one  stroke  from  —  35°  to  100°,  but,  to 
economize  space,  is  broken  at  0°,  so  that  at  this  temperature 
the  progress  of  the  curve  cannot  be  determined  so  satisfactorily 
as  it  can  within  the  separate  portions  below  0°  and  above  0°. 
From  the  way  in  which  the  differences  occur  in  the  foregoing 
table,  it  would  seem  that  the  value  4.60  mm.  taken  iorp  atO°  is  a 

little  too  great,  since  if  that  were  so  the  values  of  Ap  (s—a)  - 

for  the  temperatures  just  under  0°  would  come  out  too  small, 
and  for  those  just  over  0°  too  large.  Above  100°  the  values  of 
this  expression  do  not  diminish  so  regularly  as  between  35°  and 
95°;  and  yet  they  show,  at  least  in  general,  a  corresponding 
progress  ;  and  especially  if  we  use  a  graphic  representation,  we 
find  that  the  curve,  which  within  that  interval  almost  exactly 
joins  the  successive  points  determined  by  the  numbers  contained 
in  the  table,  may  be  produced  beyond  that  interval  even  to  230° 
quite  naturally,  so  that  these  points  are  evenly  distributed  on 
both  sides  of  it. 

Within  the  range  of  the  table  the  progress  of  the  curve  can 
be  represented  with  fair  accuracy  by  an  equation  of  the  form 

(20)  Ap  (s  -  a)  -—  =  m  -  ne", 


where  e  is  the  base  of  the  natural  logarithms,  and  m,  n,  and  k 
are  constants.  If  these  constants  are  calculated  from  the  values 
which  the  curve  gives  for  45°,  125°,  and  205°,  we  obtain  ; 

(26fl)  »i=31.549,  7*  =  1.0486,  £=0.007138, 


MKMOIRS    ON 


and  if  for  convenience  we  introduce  the  Briggsian  logarithms, 
we  obtain 

0.0206  +  0.003  100  /. 


Log  [31.549-  Ap(s-v)  -^-1  =0. 
L  «.+  *J 


The  numbers  contained  in  the  third  column  are  calculated 
from  this  equation,  and  in  the  fourth  are  given  the  differences 
between  these  numbers  and  those  in  the  second  column. 

From  the  foregoing  we  may  easily  deduce  a  formula  by 
which  we  can  more  definitely  determine  the  way  in  which  the 
behavior  of  a  vapor  departs  from  the  M.  and  G.  law.  By  as- 
suming this  law  to  hold,  and  denoting  by  pxa  the  value  of  ps 
at  0°,  we  would  have  from  (20), 


p»o      a 
and    would   have,   therefore,  for    the    differential    coefficient 

—  {^—)  a  constant  quantity  —  namely,  the  well-known  coeffi- 
cient of  expansion  -  =  0.003665.  Instead  of  this  we  have  from 
(26),  if  we  simply  replace  *  —  a  by  *,  the  equation  : 


m—  n 


pg0-  m_M 
and  hence  follows  : 

/8fl)        d.(pL\-*.>± 

dt  \psj~a 

The  differential  coefficient  is,  therefore,  not  a  constant,  but  a 
function  of  the  temperature  which  diminishes  as  the  tempera- 
ture increases.  If  we  substitute  the  numerical  values  of  ///.  //. 
and  k,  given  in  (26«),  we  obtain,  among  others,  the  following 
values  for  this  function  : 

IV 


1 

itta 

t 

i(S) 

t 

5(£) 

ixg. 
0 
10 
20 
80 
40 
50 
60 

0.00842 
0.00888 
0.00884 
0.00839 
0.00825 
0.00819 
0.00814 

i>,, 
70 
80 

'.H. 

100 
110 
120 
130 

0.00807 
0.00800 
0.00298 
0.00385 
0.00276 
0.00266 
0.00256 

iiiiiii? 

000244 
•160 

0.<*' 

0.001-7 
0108 

o.omr.t 

100 


THE    SECOND    LAW    OF    THERMODYNAMICS 

It  appears  from  this  table  that  at  low  temperatures  the  de- 
partures from  the  M.  and  G.  law  are  only  slight,  but  that  at 
higher  temperatures — for  example,  at  100°,  and  upwards — they 
can  no  longer  be  neglected. 

It  may  appear  at  first  sight  remarkable  that  the  values  found 

for  4-  ( — )  are  smaller  than  0.003665,  since  we  know  that  in 
dt  \psj 

the  case  of  gases,  especially  of  those,  like  carbonic  acid  and 
sulphurous  acid,  which  deviate  most  widely  from  the  M.  and 
G.  law,  the  coefficient  of  expansion  is  not  smaller,  but  greater, 
than  that  number.  We  are  not,  however,  justified  in  making 
an  immediate  comparison  between  the  differential  coefficients 
which  we  have  just  determined  and  the  coefficient  of  expan- 
sion in  the  ordinary  sense  of  the  words,  which  relate  to  the 
increase  of  volume  at  constant  pressure,  nor  yet  with  the  num- 
ber obtained  by  keeping  the  volume  constant  during  the  heating 
process,  and  then  observing  the  increase  in  the  expansive  force. 
We  are  dealing  here  with  a  third  special  case  of  the  general 

differential  coefficient  -7-1  — ) — namely,  with  that  which  arises 
dt  \p8j 

when,  as  the  heating  goes  on,  the  pressure  increases  in  the 
same  proportion  as  it  does  with  water  vapor  when  it  is  kept  at 
its  maximum  density ;  and  we  must  consider  carbonic  acid  in 
these  relations  if  we  wish  to  institute  a  comparison. 

Water  vapor  has  a  tension  of  lm  at  about  108°,  and  of  2m  at 
129£°.  We  will,  therefore,  examine  the  behavior  of  carbonic 
acid  if  it  is  heated  by  21£°,  and  if  the  pressure  upon  it  is  at 
the  same  time  increased  from  lm  to  2m.  According  to  Reg- 
nault  *  the  coefficient  of  expansion  of  carbonic  acid  at  the  con- 
stant pressure  760mra  is  0.003710,  and  at  the  pressure  2520ram  is 
0.003846.  For  a  pressure  of  1500mm  (the  mean  between  lm  and 
2m),  if  we  consider  the  increase  of  the  coefficient  of  expansion 
as  proportional  to  the  increase  of  pressure,  we  obtain  the  value 
0.003767.  If  carbonic  acid  were  heated  at  this  mean  pressure 

from  0°  to  21£°,  the  magnitude  —  —  would  increase  from  1  to 

1  +  0.003767x21.5  =  1.08099.  Now  from  others  of  Regnault'£ 
researches  f  it  is  known  that  if  carbonic  acid,  taken  at  a  tem- 
perature near  0°  under  the  pressure  lm,  is  subjected  to  the 

*  Mem.  de  PAcad.,  Mem.  i.  f  Ibid. ,  Mem.  vi. 

101 


MEMOIRS    ON 

pressure  1.98292m,  the  magnitude  ps  decreases  in  the  ratio  of 
1  :  0.99146  ;  so  that  for  an  increase  of  pressure  from  lm  to  2m 
there  would  be  a  decrease  of  this  magnitude  in  the  ratio  of 
1  :  0.99131.  If,  now,  both  operations  were  performed  at  once  — 
that  is,  the  elevation  of  temperature  from  0°  to  21$°  and  the  in- 

crease in  pressure  from  lm  to  2m  —  the  magnitude  —  would  in- 

crease nearly  from  1  to  1.08099x0.99131  =  1.071596,  and  lienee 
we  obtain  for  the  mean  value  of  the  differential  coefficient 


=0.00333. 


It  appears,  therefore,  that  in  the  case  now  under  consideration.  :v 
value  is  obtained  for  carbonic  acid  which  is  less  than  0.003665, 
and  therefore  a  similar  result  for  a  vapor  at  its  maximum  deit*i/i/ 
should  not  be  considered  at  all  improbable. 

If,  on  the  other  hand,  we  were  to  determine  the  real  coefficient 
of  expansion  of  the  vapor  —  that  is,  the  number  which  expresses 
by  how  much  a  quantity  of  vapor  expands  if  it  is  taken  at  a 
certain  temperature  at  its  maximum  density,  and  then  removed 
from  the  water  and  heated  under  constant  pressure—  we  should 
certainly  obtain  a  value  which  would  be  greater,  and  perhaps 
considerably  greater,  than  o.i"i;;r,t;:,. 

From  equation  (26)  we  easily  obtain  tin-  rcla/in-  volumes  of 
a  unit  of  weight  of  vapor  at  its  maximum  density  for  <lilT<  rent 
temperatures,  referred  to  the  volume  at  some  definite  temper- 
ature. In  order  to  calculate  the  n/isnliite  volumes  from  these 
with  sufficient  precision,  wo  must  know  the  value  of  the  constant 
A  with  greater  accuracy  than  is  as  yet  the  case. 

The  question  now  arises  whether  any  one  volume  can  l>e 
assigned  with  sufficient  accuracy  to  permit  its  use  as  a  starting- 
point  in  the  calculation  of  the  other  absolute  values  fn.m  the 
relative  values.  Muny  investigations  of  the  specific  weight  »f 
water  vapor  have  been  carried  out,  the  results  of  which,  ln»\v- 
cver.  are  not,  in  my  opinion,  conclusive  for  the  case  with  which 
we  are  now  dealing,  in  which  the  vapor  is  at  its  maximum 
density.  The  numbers  which  arc  ordinarily  given,  especially  tin- 
one  obtained  by  Qay-Lnssac—  O.f>235—  agree  very  well  with  the 
theoretical  value  obtained  l»v  a>-umin^  that  'I  parts  of  hydrogen 
102 


THE    SECOND    LAW    OF    THERMODYNAMICS 

and  1  part  of  oxygen  combine  to  form  2  parts  of  water  vapor — • 
that  is,  with  the  value 

2x0.069264-  1.10563 

—  =  U.D^/i. 

These  numbers,  however,  are  obtained  from  observations  which 
were  not  carried  out  at  temperatures  at  which  the  resulting 
pressure  was  equal  to  the  maximum  expansive  force,  but  at 
higher  temperatures.  In  this  condition  the  vapor  might  nearly 
conform  to  the  M.  and  G.  law,  and  the  agreement  with  the 
theoretical  value  may  thus  be  explained.  To  pass  from  this 
result  to  the  condition  of  maximum  density  by  the  use  of  the 
M.  and  G.  law  would  contradict  our  previous  conclusions,  since 
Table  IV.  shows  too  large  a  departure  from  this  law,  at  the 
temperatures  at  which  the  determination  was  made,  to  make 
such  a  use  of  the  law  possible.  Those  experiments  in  which 
the  vapor  was  observed  at  its  maximum  density  give  for  the 
most  part  larger  numbers,  and  Regnault  has  concluded  *  that 
even  at  a  temperature  a  little  over  30°,  in  the  case  in  which  the 
vapor  is  developed  in  vacuum,  a  sufficient  agreement  with  the 
theoretical  value  is  reached  only  when  the  tension  of  the  vapor 
amounts  to  no  more  than  0.8  of  that  which  corresponds  to  the 
observed  temperature  as  the  maximum.  A  definite  conclusion, 
however,  cannot  be  drawn  from  this  observation,  since  it  is 
doubtful,  as  Regnault  remarks,  whether  the  departure  is  really 
due  to  too  great  a  specific  weight  of  the  vapor  formed,  or  whether 
a  quantity  of  water  remained  condensed  on  the  walls  of  the  glass 
globe.  Other  experiments,  which  were  so  executed  that  the 
vapor  did  not  form  in  vacuum  but  saturated  a  current  of  air, 
gave  results  which  were  tolerably  free  from  any  irregularities,! 
yet  even  these  results,  important  as  they  are  in  other  relations, 
do  not  enable  us  to  form  any  definite  conclusions  as  to  the  be- 
havior of  vapor  in  a  vacuum. 

In  this  state  of  uncertainty  the  following  considerations  may 
perhaps  be  of  some  service  in  filling  the  gap.  Table  IV.  shows 
that  the  vapor  at  its  maximum  density  conforms  more  closely 
to  the  M.  and  G.  law  as  the  temperature  is  lower,  and  it  may 
hence  be  concluded  that  the  specific  weight  will  approach  the 
theoretical  value  more  nearly  at  lower  than  at  higher  temper- 
atures. If  therefore,  for  example,  we  assume  the  value  0.622 

*  Ann.  Oe  Chim.  et  de  Phys.t  III.  Ser.,  t.  xv.,  p.  148.        f  Ibid.,  p.  158  ff. 
103 


MEMOIRS    OX 

as  correct  for  0°  and  then  calculate  the  corresponding  value  <l 
for  higher  temperatures  by  the  help  of  the  following  equation 
deduced  from  (26), 

(30)  (7  =  0.622   m  ""*„.. 

m  —  nr* 

we  obtain  much  more  probable  values  than  if  we  were  to  adopt 
0.622  as  correct  for  all  temperatures.  The  following  table 
presents  some  of  these  values : 


/ 

0° 

50° 

V 

100° 

wo8 

200° 

d 

0.622 

0.631 

0.645 

0.666 

0.698 

Strictly  speaking,  we  must  go  further  than  this.  In  Table  III. 
we  see  that  the  values  of  Ap  (s  —  a)  --  -,  as  the  temperature 

falls,  approach  a  limiting  value,  which  is  not  reached  even  for 
the  lowest  temperatures  of  the  table,  and  it  is  only  for  this 
limiting  value  that  we  have  a  right,  to  assume  the  applicability 
of  the  M.  and  G.  law  and  so  set  the  specific  weight  equal  to 
0.622.  The  question  therefore  arises  what  this  liniiiinir  value 
is.  If  we  could  consider  the  formula  (26)  as  applicable  for 
temperatures  below  —15°,  we  would  have  only  to  take  the  value 
which  it  approaches  asymptotically,  m  =  31.549,  and  we  could 
then  replace  equation  (30)  by  the  equation 


(3D 


From  this  equation  we  obtain  for  the  specific  weight  at  0°  the 
value  0.643  instead  of  (U522.  and  the  other  numbers  of  the 
preceding  table  must  be  increased  in  the  same  ratio.  We  are, 
however,  not  justified  in  so  extended  an  application  of  formula 
(26),  since  it  is  only  obtained  empirically  from  the  values  given 
in  Table  HI.,  and  of  these,  those  which  relate  to  the  lowest 
temperatures  are  rather  uncertain.  We  must  therefore,  for  1  1  it- 

present,  treat  the  limiting  value  of  A  (9  —  <r)  —  —  r  as  unknown, 

and  content  ourselves  with  such  an  approximation  as  the  num- 
bers in  the  preceding  tables  warrant.     We  may.  however,  con- 
clude that  these  numbers  are  rather  too  small  than  too  great. 
104 


THE    SECOND    LAW    OF    THERMODYNAMICS 

If  we  combine  equation  (V.o)  with  equation  (III.)  deduced 
from  the  first  principle,  we  may  eliminate  A  (s  —  a),  and  obtain : 

<*>  »*-»-5T». 

By  means  of  this  equation  we  may  determine  the  magnitude  h, 
which  has  already  been  stated  to  be  negative.  If  we  set  for  c  and 
r  the  expressions  given  in  (23b)  and  (24),  and  for  a  the  number 
273,  we  obtain : 

606.    -  0. 695  /  -  0. 00002  P  -  0. 0000003 13 
(33)      *=:<UJG5~-  273  +  *  ~J 

and  hence  obtain  for  //,  among  others,  the  values : 


•0° 

50° 

VI 

100° 

-1.916 

-1.465 

-1.133 

150C 


-0.879 


200C 


-0.676 


In  a  way  similar  to  that  which  we  have  followed  in  the  case 
of  water  vapor,  we  might  apply  equation  (V.a)  to  the  vapors  of 
other  liquids  also,  and  then  compare  the  results  obtained  for 
these  different  liquids,  as  has  been  done  with  the  numbers  cal- 
culated by  Clapeyron  and  contained  in  Table  I.  "We  shall  not, 
however,  go  into  these  applications  any  further  at  present. 

We  must  now  endeavor  to  determine,  at  least  approximately, 
the  numerical  value  of  the  constant  A,  or,  what  is  more  useful, 

of  the  fraction  —  ,  that  is,  the  work  equivalent  of  tlte  unit  of  heat. 

For  this  purpose  we  can  first  use  equation  (lOa)  for  the  per- 
manent gases,  which  amounts  to  the  same  thing  as  the  method 
already  employed  by  Mayer  and  Helmholtz.  This  equation  is  : 

c'  =  c  +  AR, 

c' 
and  if  we  set  for  c  the  equivalent  expression  -p,  we  have  : 


The  value  commonly  taken  for  c'  for  atmospheric  air  from 
the  researches  of  De  Laroche  and  Berard  is  0.267,  and  for  k 
from  the  researches  of  Dulong  is  1.421.  Further,  to  determine 

R  =         "  ,  we  know  that  the  pressure  of  one  atmosphere 

a  -f-  TO 

(760mm)  on  a  square  meter  is  10333  kilogrammes,  and  that  the 
105 


MEMOIRS    ON 

volume  of  one  kilogramme  of  atmospheric  air  under  that  pr> 
and  at  the  temperature  of  the  freezing-point  =0.7733  cubic 
meters.     Hence  follows : 

10333. 0.7733 

273 
and  consequently 

±  _  1.421.29.26  _ 
,4-0.421.0.267- 

that  is,  by  the  expenditure  of  a  unit  of  heat  (that  quantity  of 
heat  which  will  raise  the  temperature  of  1  kilogramme  of  water 
from  0°  to  1°)  370  kilogrammes  can  be  lifted  to  the  height  of 
lm.  Little  confidence  can  be  placed  in  this  number,  on  account 
of  the  uncertainty  of  the  numbers  0.267  and  1.421.  Holt/maim 
gives  as  the  limits,  between  which  he  is  in  doubt.  :M:J  ami  I  .".i. 
We  may  further  use  the  equation  (V.a)  developed  for  vapors. 
If  we  wish  to  apply  it  to  water  vapor,  we  can  use  the  determi- 
nations given  in  the  former  part  of  our  work,  whose  result  is 
expressed  in  equation  (26).  If  we  choose  in  this  equation  the 
temperature  100°,  for  example,  and  set  for  p  the  corresponding 
pressure  of  1  atmosphere  =  10333  kilogrammes,  we  obtain : 

(35)  1=257  (*-*). 

If  we  now  use  Gay-Lussac's  value  of  the  specific  weight  of 
water  vapor,  0.6235,  we  obtain  s= 1.699,  and  hence, 

1=43, 

Similar  values  are  given  by  the  use  of  the  numbers  contained 
in  Table  I.,  which  Clapeyron  and  Thomson  have  calculated 
for  C  from  equation  (V.).  For  if  we  consider  these  as  the 
values  of  A  (a+t)  for  the  temperatures  corresponding  to  them, 

we  obtain  for-j  a  set  of  values  which  lie  between  416  and  462. 
A 

It  has  already  been  mentioned  that  the  specific  weight  of 
water  vapor  given  by  Gay-Lussac  is  probably  somewhat  too 
small  for  the  case  where  the  vapor  is  at  its  maximum  density. 
The  same  may  be  said  of  most  of  the  specific  weights  which  an> 
ordinarily  given  for  other  vapors.  We  must  therefore  con- 
clude that  the  values  of  -j  calculated  from  them  are  for  tin- 

A 

most  part  a  little  too  great.     If  we  take  for  water  vapor  the 
100 


THE    SECOND    LAW    OF    THERMODYNAMICS 

number  0.645  given  in  Table  V.,  from  which  s=1.638,  we 
obtain  i 

z=421- 

This  value  is  also  perhaps  a  little,  but  probably  not  much, 
too  great.  We  may  therefore  conclude,  since  this  result 
should  be  given  the  preference  over  that  obtained  from  atmos- 
pheric air,  that  the  work  equivalent  of  the  unit  of  heat  is  the 
lifting  of  something  over  400  kilogrammes  to  the  height  0/lm. 

We  may  now  compare  with  this  theoretical  result  those  which 
Joule  obtained  in  very  different  ways  by  direct  observation. 
Joule  obtained  from  the  heat  produced  by  magneto-electricity, 

1=460;' 

from  the  quantity  of  heat  which  atmospheric  air  absorbs  during 
its  expansion,  l 

^=438,t 

and  as  a  mean  of  a  large  number  of  experiments,  in  which  the 
heat  produced  by  friction  of  water,  of  mercury,  and  of  cast- 
iron,  was  observed,  -i 

Z=«5.t 

The  agreement  of  these  three  numbers,  in  spite  of  the  diffi- 
culty of  the  experiments,  leaves  really  no  further  doubt  of  the 
correctness  of  the  fundamental  principle  of  the  equivalence  of 
heat  and  work,  and  their  agreement  with  the  number  421  con- 
firms in  a  similar  way  the  correctness  of  Carnot's  principle,  in 
the  form  which  it  takes  when  combined  with  the  first  principle. 


BIOGRAPHICAL  SKETCH 

RUDOLF  JULIUS  EMAJTUEL  CLAUSIUS  was  born  on  January 
2,  1822,  at  Coslin,  in  Pomerania.  He  was  educated  at  Berlin, 
and  became  Privat-docent  in  the  University  of  Berlin  and  In- 
structor in  Physics  at  the  School  of  Artillery.  In  1855  he  was 
appointed  to  the  Professorship  of  Physics  in  the  Polytechnic 
School  at  Zurich,  and  in  1857  he  was  appointed  to  a  similar 

*  Phil.  Mag.,  xxiii.,  p.  441.  The  number,  given  in  English  units,  is  re- 
duced to  French  units. 

f  Ibid.,  xxvi ,  p.  381.  \  Ibid.,  xxxv.,  p.  534. 

107 


THE    SECOND    LAW    OF    THERMODYNAMICS 

position  in  the  University  of  Zurich.  In  1869  he  was  appointed 
1'rufessor  of  Physics  in  the  University  of  Bonn,  where  lu-  re- 
mained until  his  death,  on  August  24,  1888. 

Clausius  was  a  prolific  investigator  and  writer  on  physical 
subjects.  The  line  of  thought  suggested  by  the  discoveries  in 
heat  contained  in  the  memoir  given  in  this  volume  was  fol- 
lowed out  by  him  in  a  series  of  papers  on  the  thermodynamic 
properties  of  bodies  and  on  the  general  theory  of  thermody- 
namics. These  papers  were  collected  and  published  in  a  volume 
in  1864;  and  ten  years  later  he  recast  these  papers  and  others 
which  had  appeared  after  the  collection  was  first  published  into 
a  systematic  treatise  on  the  mechanical  theory  of  heat.  The 
concept  of  the  entropy,  which  Clausing  introduced  and  de- 
veloped, is  the  most  important  single  contribution  made  by 
him  to  science. 

Clausius's  investigations  also  extended  into  radiant  heat,  in 
connection  with  which  he  proved  that  radiance  also  conforms 
to  the  second  law  of  thermodynamics.  Clausius  was  the  first 
to  apply  the  doctrine  of  probabilities,  in  any  systematic  way,  to 
the  kinetic  theory  of  gases  ;  and  by  so  doing  he  laid  the  foun- 
dations for  the  brilliant  applications  of  that  doctrine  to  the 
kinetic  theories  which  have  been  made  by  Maxwell  and  Boltz- 
mann.  He  also  contributed  something  to  the  theory  of  elec- 
tricity. His  writings  are  characterized  by  simplicity  of  form 
and  profundity  of  thought.  They  deal  much  with  fundamental 
questions,  but  by  such  direct  and  simple  methods  that  the 
ideas  under  discussion  are  rarely  obscured  by  the  difficulties  of 
the  analysis. 


ON  THE  DYNAMICAL  THEORY  OF  HEAT, 
WITH  NUMERICAL  RESULTS  DEDUCED 
FROM   MR.  JOULE'S   EQUIVALENT 
OF    A    THERMAL    UNIT,    AND 
M.  REGNAULT'S  OBSERVA- 
TIONS  ON    STEAM 

BY 

WILLIAM  THOMSON  (LORD  KELVIN) 


(Transactions  of  the  Royal  Society  of  Edinburgh,  March,  1851 ;  Philosopliical 
Magazine,  iv.,  1852;  Mathematical  and  Physical  Papers,  vol.  i.,  p.  174) 


CONTENTS 

MM 

Introductory  Notice Ill 

Fundamental  Principle* 114 

Carnot'i  Cycle 116 

Carnot't  Function 128 

Jt»  Detenni 'nation 183 

Tktrmoilyiuiinic  Relation* . .  186 


ON  THE  DYNAMICAL  THEORY  OF  HEAT 

BY 

WILLIAM  THOMSON 


INTRODUCTORY  NOTICE 

1.  SIR  HUMPHRY  DAVY,  by  his  experiment  of  melting  two 
pieces  of  ice  by  rubbing  them  together,  established  the  follow- 
ing proposition :    "  The  phenomena  of  repulsion  are  not  de- 
pendent  on  a  peculiar  elastic   fluid   for  their  existence,  or 
caloric  does  not  exist."     And  he  concludes  that  heat  consists 
of  a  motion  excited  among  the  particles  of  bodies.     "  To  dis- 
tinguish this  motion  from  others,  and  to  signify  the  cause  of 
onr  sensation  of  heat,"  and  of  the  expansion  or  expansive  press- 
ure produced  in  matter  by  heat,  "the  name  repulsive  motion 
has  been  adopted."* 

2.  The  dynamical  theory  of  heat,  thus  established  by  Sir 
Humphry  Davy,  is  extended  to  radiant  heat  by  the  discovery 
of  phenomena,  especially  those  of  the  polarization  of  radiant 
heat,  which  render  it  excessively  probable  that  heat  propagated 
through  "vacant  space,"  or  through  diathermanic  substances, 
consists  of  waves  of  transverse  vibrations  in  an  all-pervading 
medium. 

3.  The  recent  discoveries  made  by  Mayer  and  Joule, f  of  the 

*  From  Davy's  first  work,  entitled  An  Essay  on  H&it,  Light,  and  the  Com- 
binations of  Light,  published  in  1799,  in  "  Contributions  to  Physical  and 
Medical  Knowledge,  principally  from  the  West  of  England,  collected  by 
Thomas  Beddoes,  M.D .,"  and  republished  in  Dr.  Davy's  edition  of  his 
brother's  collected  works,  vol.  ii.,  Loud.,  1836. 

f  In  May,  1842,  Mayer  announced  in  the  Annalen  of  Wohler  and  Liebig, 
that  he  had  raised  the  temperature  of  water  from  12°  to  13°  Cent,  by  agi- 
tating it.  In  August,  1843,  Joule  announced  to  the  British  Association 
111 


MEMOIRS    ON 

generation  of  heat  through  the  friction  of  fluids  in  motion,  and 
by  the  magneto-electric  excitation  of  galvanic  currents,  would 
either  of  them  be  sufficient  to  demonstrate  the  immateriality 
of  heat ;  and  would  so  afford,  if  required,  a  perfect  continua- 
tion of  Sir  Humphry  Davy's  views. 

4.  Considering  it  as  thus  established,  that  heat  is  not  a  sub- 
stance, but  a  dynamical  form  of  mechanical  effect,  we  jn-rcrive 
that  there  must  be  an  equivalence  between  mechanical  work 
and  heat,  as  between  cause  and  effect.  The  first  published 
statement  of  this  principle  appears  to  be  in  Mayer's  J: 
knnyen  ilfar  '//'•  AV<>/"/V  <l<  r  /////"/<///,•//  \<ttiu\*  which  contains 
some  correct  views  regard inir  the  mutual  convertibility  of  heat 
and  mechanical  effect,  along  with  a  false  analogy  between  the 
approach  of  a  weight  to  the  earth  and  a  diminution  of  the  vol- 
ume of  a  continuous  substance,  on  which  an  attempt  is  founded 
to  find  numerically  the  mechanical  equivalent  of  a  given  quan- 
tity of  heat.  In  a  paper  published  about  fourteen  months 
later,  "On  the  Calorific  Effects  of  Magneto-Electricity  and  the 
Mechanical  Value  of  IIeat,"f  Mr.  Joule,  of  Manchester.  «  \- 
presses  very  distinctly  the  consequences  regarding  the  mutual 
convertibility  of  heat  and  mechanical  effect  which  follow  from 
the  fact  that  heat  is  not  a  substance  but  a  state  of  motion  ; 
and  investigates  on  unquestionable  principles  the  >%al»olutc 
numerical  relations/'  according  to  which  heat  is  connected 
with  mechanical  power;  verifying  experimentally,  that  when- 
ever heat  is  generated  from  purely  mechanical  action,  and  no 
other  effect  produced,  whether  it  be  by  means  of  the  friction 
of  fluids  or  by  the  magneto-electric  excitation  of  galvanic  cur- 
rents, the  same  quantity  is  generated  by  the  same  amount  of 
work  spent;  and  determining  the  actual  amount  of  work,  in 
foot-pounds,  required  to  generate  a  unit  of  heat,  whirh  he 
calls  "the  mechanical  equivalent  of  heat."  Since  the  publica- 

"Tbat  heat  is  evolved  by  the  passage  of  water  through  narrow  uiU>s;' 
and  that  he  hat]  "obtained  one  degree  of  heat  per  pound  of  water  fn.m  a 
UK  <  iiaiiic:il  force  capable  of  raising  770  pounds  to  the  height  of  one  f.-.t  ." 
and  that  heat  is  generated  when  work  is  spent  in  turning  a  magneto-elec- 
tric machine,  or  an  electro  •  magnetic  engine.  (See  his  paper  "On  the 
Calorific  Effects  of  Magm-io  KI«-«  ttiriiy,  and  on  tli«-  M<-<  li.mirul  Value  of 
Heat."— Phil.  Mag.,  vol.  xxiii..  1848.) 

*  AnnaUn  of  Wol.ler  and  Liebig.  May.  1842. 

f  British  Association,  August,  1848;  and  Phil.  Mag.,  September,  1848. 
Ill 


THE    SECOND    LAW    OF    THERMODYNAMICS 

tion  of  that  paper,  Mr.  Joule  has  made  numerous  series  of  ex- 
periments for  determining  with  as  much  accuracy  as  possible 
the  mechanical  equivalent  of  heat  so  defined,  and  has  given 
accounts  of  them  in  various  communications  to  the  British 
Association,  to  the  Philosophical  Magazine,  to  the  Royal  So- 
ciety, and  to  the  French  Institute. 

5.  Important  contributions  to  the  dynamical  theory  of  heat 
have  recently  been  made  by  Rankine  and  Clausius  ;  who,  by 
mathematical  reasoning  analogous  to  Carnot's  on  the  motive 
power  of  heat,  but  founded  on  an  axiom  contrary  to  his  funda- 
mental axiom,  have  arrived  at  some  remarkable  conclusions. 
The  researches  of  these  authors  have  been  published  in  the 
Transactions  of  this  Society,  and  in  Poggendorff's  Annalen, 
during  the  past  year  ;  and  they  are  more  particularly  referred 
to  below  in  connection  with  corresponding  parts  of  the  investi- 
gations at  present  laid  before  the  Royal  Society. 

6.  The  object  of  the  present  paper  is  threefold  : 

(1)  To  show  what  modifications  of  the  conclusions  arrived 
at  by  Carnot,  and  by  others  who  have  followed  his  peculiar 
mode  of  reasoning  regarding  the  motive  power  of  heat,  must 
be  made  when  the  hypothesis  of  the  dynamical  theory,  con- 
trary as  it  is  to  Carnot's  fundamental  hypothesis,  is  adopted. 

(2)  To  point  out  the  significance  in  the  dynamical  theory, 
of  the  numerical  results  deduced  from  Regnault's  observations 
on  steam,  and  communicated  about  two  years  ago  to  the  So- 
ciety, with  an  account  of  Carnot's  theory,  by  the  author  of 
the  present  paper ;  and  to  show  that  by  taking  these  numbers 
(subject  to  correction  when  accurate  experimental  data  regard- 
ing the  density  of  saturated  steam  shall  have  been  afforded), 
in  connection  with  Joule's  mechanical  equivalent  of  a  ther- 
mal unit,  a  complete  theory  of  the   motive  power  of  heat, 
within  the  temperature  limits  of  the  experimental  data,  is  ob- 
tained. 

(3)  To  point  out  some  remarkable  relations  connecting  the 
physical  properties  of  all  substances,  established  by  reasoning- 
analogous  to  that  of  Carnot,  but  founded  in  part  on  the  con- 
trary principle  of  the  dynamical  theory. 


MEMOIRS    ON 

PART  I 
Fundamental  Principle*  in  the  Tlteory  of  tlie  Motive  Power  of 


7.  According  to  an  obvious  principle,  first  introduced,  how- 
ever, into  the  theory  of  the  motive  power  of  heat  by  Carnot. 
mechanical  effect  produced  in  any  process  cannot  be  said  to 
have  been  derived  from  a  purely  thermal  source,  unless  at  the 
end  of  the  process  all  the  materials  used  are  in  precisely  the 
same  physical  and  mechanical  circumstances  as  they  were  at 
the  beginning.      In  some  conceivable  "  thermo-  dynamic  en- 
gines," as,  for  instance,  Faraday's  floating  magnet,  or  Barlow's 
"wheel  and  axle,"  made  to  rotate  and  perform  work  uniformly 
by  means  of  a  current  continuously  excited  by  heat  communi- 
cated to  two  metals  in  contact,  or  the  thermo-electric  rotatory 
apparatus  devised  by  Marsh,  which  has  been  actually  construct- 
ed, this  condition  is  fulfilled  at  every  instant.     On  the  other 
hand,  in  all  thermo -dynamic  engines,  founded  on  electrical 
agency,  in  which  discontinuous  galvanic  currents,  or  pieces  of 
soft  iron  in  a  variable  state  of  magnetization,  are  used,  and  in 
all  engines  founded  on  the  alternate  expansions  and  contrac- 
tions of  media,  there  are  really  alterations  in  the  condition  of 
materials ;  but,  in  accordance  with  the  principle  stated  above, 
these  alterations  must  be  strictly  periodical.     In  any  such  en- 
gine the  series  of  motions  performed  during  a  period,  at  the 
end  of  which  the  materials  are  restored  to  precisely  the  same 
condition  as  that  in  which  they  existed  at  the  beginning,  con- 
stitutes what  will  be  called  a  complete  cycle  of  its  operations. 
Whenever  in  what  follows,  the  tmrk  ilnnc  or  tin   iiit'r/ttinir>tl  tf- 
fect  produced  by  a  thermo-dynamic  engine  is  mentioned  with- 
out qualification,  it  must  be  understood  that   the  mechanical 
effect  produced,  either  in  a  non-varying  engine,  or  in  a  com- 
plete cycle,  or  any  number  of  complete  cycles  of  a  periodical 
engine,  is  meant. 

8.  The  source  of  heat  will  always  be  supposed  to  be  a  hot 
body  at  a  given   constant  temperature   put    in  contact  with 
some  part  of  the  engine ;  and  when  any  part  of  the  engine  is 
to  be  kept  from  rising  in  temperature  (which  can  only  he  done 
by  drawing  off  whatever  heat  is  deposited  in  it),  this  will  be 
supposed  to  be  done  by  putting  a  cold  body,  which  will  be 

114 


THE    SECOND    LAW    OF    THERMODYNAMICS 

called  the  refrigerator,  at  a  given  constant  temperature  in  con- 
tact with  it. 

9.  The  whole  theory  of  the  motive  power  of  heat  is  founded 
on  the  two  following  propositions,  due  respectively  to  Joule, 
and  to  Carnot  aild  Clausius. 

PROP.  I.  (Joule). — When  equal  quantities  of  mechanical  ef- 
fect are  produced  by  any  means  whatever  from  purely  thermal 
sources,  or  lost  in  purely  thermal  effects,  equal  quantities  of 
heat  are  put  out  of  existence  or  are  generated. 

PROP.  II.  (Carnot  and  Clausius). — If  an  engine  be  such  that, 
when  it  is  worked  backwards,  the  physical  and  mechanical 
agencies  in  every  part  of  its  motions  are  all  reversed,  it  pro- 
duces as  much  median ical  effect  as  can  be  produced  by  any 
thermo-dynamic  engine,  with  the  same  temperatures  of  source 
and  refrigerator,  from  a  given  quantity  of  heat. 

10.  The  former  proposition  is  shown  to  be  included  in  the 
general  "principle  of  mechanical  effect,"  and  is  so  established 
beyond  all  doubt  by  the  following  demonstration. 

11.  By  whatever  direct  effect  the  heat  gained  or  lost  by  a 
body  in  any  conceivable  circumstances  is  tested,  the  measure- 
ment of  its  quantity  may  always  be  founded  on  a  determination 
of  the  quantity  of  some  standard  substance,  which  it  or  any 
equal  quantity  of  heat  could  raise  from  one  standard  temper- 
ature to  another ;  the  test  of  equality  between  two  quantities 
of  heat  being  their  capability  of  raising  equal  quantities  of  any 
substance  from  any  temperature  to  the  same  higher  temper- 
ature.    Now,  according  to  the  dynamical  theory  of  heat,  the 
temperature  of  a  substance  can  only  be  raised  by  working  upon 
it  in  some  way  so  as  to  produce  increased  thermal  motions 
within  it,  besides  effecting  any  modifications  in  the  mutual  dis- 
tances or  arrangements  of  its  particles  which  may  accompany  a 
change  of  temperature.     The  work  necessary  to  produce  this 
total  mechanical  effect  is  of  course  proportional  to  the  quantity 
of  the  substance  raised  from  one  standard  temperature  to  an- 
other ;  and  therefore  when  a  body,  or  a  group  of  bodies,  or  a 
machine,  parts  with  or  receives  heat,  there  is  in  reality  me- 
chanical effect  produced  from  it,  or  taken  into  it,  to  an  ex- 
tent precisely  proportional  to  the  quantity  of  heat  which  it 
emits  or  absorbs.     But  the  work  which  any  external  forces  do 
upon  it,  the  work  done  by  its  own  molecular  forces,  and  the 
amount  by  which  the  half  vis  viva  of  the  thermal  motions  of 

115 


MEMOIRS    ON 

all  its  parts  is  diminished,  must  together  be  equal  to  the  mc- 
chanical  effect  produced  from  it:  and,  consequently,  to  tin- 
mechanical  equivalent  of  the  heat  which  it  emits  (which  will 
be  positive  or  negative,  according  as  the  sum  of  those  term.-  is 
positive  or  negative).  Now  let  there  be  either  no  molecular 
change  or  alteration  of  temperature  in  any  part  of  the  body. 
or,  by  a  cycle  of  operations,  let  the  temperature  and  physical 
condition  be  restored  exactly  to  what  they  were  at  the  begin- 
ning;  the  second  and  third  of  the  three  parts  of  tin-  \\«>ik 
which  it  has  to  produce  vanish  ;  and  we  conclude  that  the  heat 
which  it  emits  or  absorbs  will  be  the  thermal  equivalent  of  the 
work  done  upon  it  by  external  forces,  or  done  by  it  against  ex- 
ternal forces;  which  is  the  proposition  to  be  pio\ed. 

12.  The  demonstration  of  the  second  proposition  is  founded 
on  the  following  axiom  : 

It  is  impossible,  by  means  of  intnnin«t,'  mult  rial  <"j<  nry,  to 
i/i-rirr  iitirliinnntl  cfft-rl  fnnn  nnij  jmrtinii  of  ////////•/•  ////  nmlimj  it 
beloir  //it-  tt'iiijH'ratnrc  of  tin-  mli/rsf  <>/  ///>•  snrmitnilimj  uljn-t*.* 

13.  To  demonstrate  the  second  proposition,  let  .  I  and  /•'  l>c 
two  thermo-dynamic  engines,  of  which  B  satisfies  the  condi- 
tions expressed  in  the  enunciation  ;  and  let,  if  possible.  .1  de- 
rive more  work  from  a  given  quantity  of  heat  than  />',  when 
their  sources  and  refrigerators  are  at  the  same  trmpcratu:  • 
spectively.     Then  on  account  of  the  condition  of  complete  re- 

/lilittf  in  all  its  operations  which  it  fulfils.  />'  m:iy  In- 
worked  backwards,  and  made  to  restore  any  quantity  of  heat  to 
its  source,  by  the  expenditure  of  the  amount  of  work  which,  by 
its  forward  action,  it  would  derive  from  the  same  quantity  of 
heat.  If,  therefore,  B  be  worked  hackuards.  and  made  t.>  re- 
store to  the  source  of  A  (which  we  may  suppose  to  be  adjust- 
able to  the  engine  B)  as  much  heat  as  has  been  drawn  from  it 
during  a  certain  period  of  the  working  of  A,  a  smaller  amount 
of  work  will  be  spent  thus  than  was  gained  by  tin-  working 
,  of  .1.  Hence,  if  such  a  series  of  operations  of  ./  forward-;  and 
of  //  backwards  be  continued,  either  alternately  or  simulta- 
neously, there  will  result  a  continued  production  of  work  with- 

*  If  this  axiom  be  denied  for  all  temp,  nitun •*.  it  would  him:  to  be 
ndmiilcil  I  hut  a  self  acting  mnrhim-  miiilit  he  set  to  work  and  produ.,   m<- 
chanical  effect  by  cooling  the  sea  or  nirth.  with  tu>  limit  hut  the  t<>i:il  I—- 
of heat  from  tbc  earth  and  sea.  or,  in  reality,  from  the  whole  n 
world. 

116 


THE    SECOND    LAW    OF    THERMODYNAMICS 

out  any  continued  abstraction  of  heat  from  the  source ;  and, 
by  Prop.  I.,  it  follows  that  there  must  be  more  heat  abstracted 
from  the  refrigerator  by  the  working  of  B  backwards  than  is 
deposited  in  it  by  A.  Now  it  is  obvious  that  A  might  be 
made  to  spend  part  of  its  work  in  working  B  backwards,  and 
the  whole  might  be  made  self-acting.  Also,  there  being  no 
heat  either  taken  from  or  given  to  the  source  of  the  whole,  all 
the  surrounding  bodies  and  space  except  the  refrigerator  might, 
without  interfering  with  any  of  the  conditions  which  have  been 
assumed,  be  made  of  the  same  temperature  as  the  source,  what- 
ever that  may  be.  We  should  thus  have  a  self-acting  machine, 
capable  of  drawing  heat  constantly  from  a  body  surrounded  by 
others  at  a  higher  temperature,  and  converting  it  into  me- 
chanical effect.  But  this  is  contrary  to  the  axiom,  and  there- 
fore we  conclude  that  the  hypothesis  that  A  derives  more 
mechanical  effect  from  the  same  quantity  of  heat  drawn  from 
the  source  than  B  is  false.  Hence  no  engine  whatever,  with 
source  and  refrigerator  at  the  same  temperatures,  can  get  more 
work  from  a  given  quantity  of  heat  introduced  than  any  en- 
gine which  satisfies  the  condition  of  reversibility,  which  was  to 
be  proved. 

14.  This  proposition  was  first  enunciated  by  Carnot,  being 
the  expression  of  his  criterion  of  a  perfect  thermo  -  dynamic 
engine.*  He  proved  it  by  demonstrating  that  a  negation  of 
it  would  require  the  admission  that  there  might  be  a  self- 
acting  machine  constructed  which  would  produce  mechani- 
cal effect  indefinitely,  without  any  source  either  in  heat  or  the 
consumption  of  materials,  or  any  other  physical  agency ;  but  this 
demonstration  involves,  fundamentally,  the  assumption  that, 
in  "'a  complete  cycle  of  operations,"  the  medium  parts  with 
exactly  the  same  quantity  of  heat  as  it  receives.  A  very  strong 
expression  of  doubt  regarding  the  truth  of  this  assumption,  as 
a  universal  principle,  is  given  by  Carnot  himself  ;f  and  that  it 
is  false,  where  mechanical  work  is,  on  the  whole,  either  gained 
or  spent  in  the  operations,  may  (as  I  have  tried  to  show  above) 
be  considered  to  be  perfectly  certain.  It  must  then  be  admit- 
ted that  Carnot's  original  demonstration  utterly  fails,  but  we 
cannot  infer  that  the  proposition  concluded  is  false.  The 
truth  of  the  conclusion  appeared  to  me,  indeed,  so  probable 

*  "Account  of  Carnot's  Theory,"  §  13.  f  Ibid.,  §  6. 

117 


MEMOIRS    ON 

that  I  took  it  in  connection  with  Joule's  principle,  on  account 
of  which  Caruot's  demonstration  of  it  fails,  as  the  foundation  of 
an  investigation  of  the  motive  power  of  heat  in  air-cnuincs  or 
steam-engines  through  finite  ranges  of  tempi  -ratuiv,  and  ob- 
tained about  a  year  ago  results,  of  which  the  substance  is  ijivcn 
in  the  second  part  of  the  paper  at  present  communicated  t<> 
the  Royal  Society.  It  was  not  until  the  commencement  of  the 
present  year  that  I  found  the  demonstration  given  above,  by 
which  the  truth  of  the  proposition  is  established  upon  an  axiom 
(§  12)  which  I  think  will  be  generally  admitted.  It  i.s  with  no 
wish  to  claim  priority  that  I  make  these  statements,  as  tin- 
merit  of  first  establishing  the  proposition  upon  correct  princi- 
ples is  entirely  due  to  Clausius,  who  published  his  drnnmst  ra- 
tion of  it  in  the  month  of  May  last  year,  in  the  second  part  of 
his  paper  on  the  motive  power  of  heat.*  I  may  be  allowed  to 
add  that  I  have  given  the  demonstration  exactly  as  it  occurred 
to  me  before  I  knew  that  Clausius  had  either  enunciated  or 
demonstrated  the  proposition.  The  following  is  the  axiom  on 
which  Clausius's  demonstration  is  founded  : 

//  /x  iiii/HHtxible  for  it  .«rlf-<n-tini/  iiuicliinc.  nmn'<ti'<l  In/  tt/ti/  e.r- 
l>  null  it;/i  nnj,  /o  convey  heat  from  one  bmly  1<>  unot/irr  at  a  liiyln-r 


It  is  easily  shown  that,  although  this  and  the  axiom  I  have 
used  are  different  in  form,  either  is  a  consequence  of  the  other. 
The  reasoning  in  each  demonstration  is  strictly  analogous  i.. 
that  which  Carnot  originally  gave. 

15.  A  complete  theory  of  the  motive  power  of  heat  \\uuld 
consist  of  the  application  of  the  two  proportions  dcm..n>ir:it.-d 
above  to  every  possible  method  of  producing  mechanical  elTect 
from  thermal  agency,  f  As  yet  this  has  not  been  clone  for  tin- 
electrical  method,  as  far  as  regards  the  criterion  of  a  perfect 
engine  implied  in  the  second  proposition,  and  pmliaMy  <  ann..i 
be  done  without  certain  limitations  ;  but  tin-  application  <>!  tin- 
first  proposition  ha.-  been  very  thoroughly  investigated,  and 
\crified  experimentally  by  Mr.  Joule  in  his  researches  "On  the 

•  PogRendorfTs  Annalen,  referred  to  above. 

f  "Tin-re  i»rc  at  present  known  two.  and  only  two.  distinct  ways  in 
which  mechanical  effect  can  br  nlnnim-d  fr..m  lu-at.    One  of  these  is  by  the 
alterations  of  volume  whiHi  \»»\\,  -s  ,  -\\»  THMH  •«•  through  the  action  of  in  ni  . 
the  other  is  through  the  medium  of  electric  agency."—"  Account  <>!  (   n 
not's  Theory,  '  §  4.     (Transaction*,  vol.  xvi..  part  5.) 


THE    SECOND    LAW    OF    THERMODYNAMICS 

Calorific  Effects  of  Magneto-Electricity ;"  and  on  it  is  founded 
one  of  his  ways  of  determining  experimentally  the  mechanical 
equivalent  of  heat.  Thus  from  his  discovery  of  the  laws  of 
generation  of  heat  in  the  galvanic  circuit,*  it  follows  that  when 
mechanical  work  by  means  of  a  magneto  -  electric  machine  is 
the  source  of  the  galvanism,  the  heat  generated  in  any  given 
portion  of  the  fixed  part  of  the  circuit  is  proportional  to  the 
whole  work  spent ;  and  from  his  experimental  demonstration 
that  heat  is  developed  in  any  moving  part  of  the  circuit  at  ex- 
actly the  same  rate  as  if  it  were  at  rest,  and  traversed  by  a  cur- 
rent of  the  same  strength,  he  is  enabled  to  conclude  : 

(1)  That  heat  may  be  created  by  working  a  magneto-electric 
machine. 

(2)  That  if  the  current  excited  be  not  allowed  to  produce 
any  other  than  thermal  effects,  the  total  quantity  of  heat  pro- 
duced is  in  all  circumstances  exactly  proportional  to  the  quan- 
tity of  work  spent. 

16.  Again,  the  admirable  discovery  of  Peltier,  that  cold  is 
produced  by  an  electrical  current  passing  from  bismuth  to  anti- 
mony, is  referred  to  by  Joule,  f  as  showing  how  it  may  be  proved 
that,  when  an  electrical  current  is  continuously  produced  from  a 

*  That,  in  a  given  fixed  part  of  the  circuit,  the  heat  evolved  in  a  given 
time  is  proportional  to  the  square  of  the  strength  of  the  current,  and  for 
different  fixed  parts,  with  the  same  strength  of  current,  the  quantities  of 
heat  evolved  in  equal  times  are  as  the  resistances.  A  paper  by  Mr.  Joule, 
containing  demonstrations  of  these  laws,  and  of  others  on  the  relations  of 
the  chemical  and  thermal  agencies  concerned,  was  communicated  to  the 
Royal  Society  on  the  17th  of  December,  1840,  but  was  not  published  in  the 
Transactions.  (See  abstract  containing  a  statement  of  the  laws  quoted 
above,  in  the  Philosophical  Magazine,  vol.  xviii.,  p.  308.)  It  was  published 
in  the  Philosophical  Magazine  in  October,  1841  (vol.  xix.,  p.  260). 

f  [Note  of  March  20,  1852,  added  in  Phil.  Mag.  reprint.  In  the  intro- 
duction to  his  paper  "On  the  Calorific  Effects  of  Magneto-Electricity," 
etc.,  Phil.  Mag.,  1843. 

I  take  this  opportunity  of  mentioning  that  I  have  only  recently  become 
acquainted  with  Helmholtz's  admirable  treatise  on  the  principle  of  mechani- 
cal effect  ( Ueber  die  Erhaltung  der  Kraft,  von  Dr.  H.  Helmholtz.  Berlin. 
G.  Reimer,  1847),  having  seen  it  for  the  first  time  on  the  20th  of  January 
of  this  year  ;  and  that  I  should  have  had  occasion  to  refer  to  it  on  this,  and 
on  numerous  other  points  of  the  dynamical  theory  of  heat,  the  mechanical 
theory  of  electrolysis,  the  theory  of  electro -magnetic  induction,  and  the 
mechanical  theory  of  thermo-electric  currents,  in  various  papers  communi- 
cated to  the  Royal  Society  of  Edinburgh,  and  to  this  Magazine,  had  I  been 
acquainted  with  it  in  time.— W.  T.,  March  20,  1852.] 
119 


MI: MO i us  ox 

purely  thermal  source,  the  quantities  of  heat  evolved  electri- 
cally in  the  different  homogeneous  parts  of  the  circuit  arc  nnlv 
compensations  for  a  loss  from  the  junctions  of  the  different 
nu'tals,  or  that,  when  the  effect  of  the  current  is  entirely  ther- 
mal, there  must  be  just  as  much  heat  emitted  from  the  pans 
not  affected  by  the  source  as  is  taken  from  the  soun •«-. 

17.  Lastly,*  when  a  current  produced  by  thermal  agency  is 
made  to  work  an  engine  and  produce  mechanical  effect,  there 
will  be  less  heat  emitted  from  the  parts  of  the  circuit  not  af- 
fected by  the  source  than  is  taken  in  from  the  source,  1>\  an 
amount  precisely  equivalent  to  tin-  mechanical  effect  produced  ; 
since  Joule  demonstrates  experimentally  that  a  current  from 
any  kind  of  source  driving  an  engine,  produces  in  the  enirin.- 
just  as  much  less  heat  than  it  would  produce  in  a  fixed  win- 
exercising  the  same  resistance  as  is  equivalent  to  the  mechani- 
cal effect  produced  by  the  engine. 

18.  The  quality  of   thermal   effects,  resulting  from   equal 
causes  through  very  different  means,  is  beautifully  illustrate'! 

*  This  reasoning  was  suggested  to  me  by  the  following  pasture  eon. 
tained  in  a  letter  which  I  received  from  Mr.  Joule  on  the  8ih  of  July,  is  1 7. 
"In  Peltier's  experiment  on  cold  produced  at  the  bismuth  ami  aniimony 
solder,  we  have  an  instance  of  the  conversion  of  heat  into  tin-  mechanical 
force  of  the  current."  which  must  have  been  meant  as  an  answer  to  a  re- 
murk  I  had  made,  that  no  evidence  could  be  adduced  to  show  that  In  MI  is 
ever  put  out  of  existence.  I  now  fully  admit  the  force  of  that  answer  ;  hut 
it  would  require  a  proof  that  there  is  more  heat  put  out  of  existence  at  iln> 
heated  soldering  [or  in  this  and  other  parts  of  the  circuii]  than  is  < 
at  the  eold  soldering  [and  the  remainder  of  the  circuit,  when  a  machine  is 
driven  by  the  current]  to  make  the  "evidence"  be  rr/«  riim  nf<il.  That 
this  is  the  case  I  think  in  certain,  because  the  statements  <>f  i  H>  in  the  tr\i 
lire  demonstrated  consequences  of  the  first  fundamental  ju«.|M»itinn ;  hut 
it  is  Mill  to  he  remarked  that  neither  in  this  nor  in  any  other  case  of  the 
production  of  mechanical  effect  from  purely  thermal  agency,  has  :!.' 
ing  to  exist  of  an  equivalent  quantity  i>f  heat  l»een  dem.  .titrated  otherwise 
than  theoretically.  It  would  I*  a  very  great  step  in  the  e\pei  inn -ntal  illus 
t  rat  ion  (or  r,  ririr<iti»n.  for  those  who  consider  such  to  I..  f  the 

dynamical  theory  of  heat,  to  actually  show  in  any  one  case  a  loss  of  heat  ; 
and  il  might  I*'  done  by  operating  through  a  ffMTJ   e..n-idei:,)ile  ran.je  of 
temperature*  with  a  good  air-engine  or  steam  engine,  not  allowed  to  waste 
rk  in  friction      As  will  IK?  seen  in  Tart  II    of  this  paper,  no  rxpeii 
mi  tit  of  iinv  kind  could  show  a  considerable  loss  of  heat  without  employ 
ing  bodies  differing  considerably  in  temperaiiin-  .  for  n 
mil'  h  ta  .098.  or  alxiiit  one  tenth  of  the  whole  heat  n-ed.  if  the  temperature 
of  all  the  bodies  used  be  between  0°  and  80°  (Ynt 
130 


THE    SECOND    LAW    OF    THERMODYNAMICS 

by  the  following  statement,  drawn  from  Mr.  Joule's  paper  on 
magneto-electricity.  * 

Let  there  be  three  equal  and  similar  galvanic  batteries  fur- 
nished with  equal  and  similar  electrodes  ;  let  A}  and  2?,  be  the 
terminations  of  the  electrodes  (or  wires  connected  with  the  two 
poles)  of  the  first  battery,  A2  and  Z?2  the  terminations  of  the 
corresponding  electrodes  of  the  second,  and  A3  and  B3  of  the 
third  battery.  Let  A}  and  B}  be  connected  with  the  extremi- 
ties of  a  long  fixed  wire  ;  let  A2  and  £z  be  connected  with  the 
"poles"  of  an  electrolytic  apparatus  for  the  decomposition  of 
water  ;  and  let  A3  and  B3  be  connected  with  the  poles  (or  ports 
as  they  might  be  called)  of  an  electro-magnetic  engine.  Then 
if  the  length  of  the  wire  between  A}  and  /?,,  and  the  speed  of 
the  engine  between  A3  an*d  B3,  be  so  adjusted  that  the  strength 
of  the  current  (which  for  simplicity  we  may  suppose  to  be  con- 
tinuous and  perfectly  uniform  in  each  case)  may  be  the  same  in 
the  three  circuits,  there  will  be  more  heat  given  out  in  any 
time  in  the  wire  between  A^  and  J5,  than  in  the  electrolytic  ap- 
paratus between  A2  and  #2,  or  the  working  engine  between  A3 
and  /?3.  But  if  the  hydrogen  were  allowed  to  burn  in  the  oxy- 
gen, within  the  electrolytic  vessel,  and  the  engine  to  waste  all 
its  work  without  producing  any  other  than  thermal  effects  (as 
it  would  do,  for  instance,  if  all  its  work  were  spent  in  continu- 
ously agitating  a  limited  fluid  mass),  the  total  heat  emitted 
would  be  precisely  the  same  in  each  of  these  two  pieces  of  ap- 
paratus as  in  the  wire  between  A{  and  Br  It  is  worthy  of  re- 
mark that  these  propositions  are  rigorously  true,  being  de- 
monstrable consequences  of  the  fundamental  principle  of  the 
dynamical  theory  of  heat,  which  have  been  discovered  by 
Joule,  and  illustrated  and  verified  most  copiously  in  his  exper- 
imental researches. 

19.  Both  the  fundamental  propositions  may  be  applied  in  a 
perfectly  rigorous  manner  to  the  second  of  the  known  meth- 
ods of  producing  mechanical  effect  from  thermal  agency.  This 
application  of  the  first  of  the  two  fundamental  propositions  has 
already  been  published  by  Rankine  and  Clausius  ;  and  that  of 
the  second,  as  Clausius  showed  in  his  published  paper,  is  sim- 

*  In  this  paper  reference  is  made  to  his  previous  paper  "On  the  Heat  of 
Electrolysis"  (published  in  vol.  vii.,  part  2,  of  the  second  series  of  the  Lit- 
erary and  Philosophical  Society  of  Manchester)  for  experimental  demon- 
stration of  those  parts  of  the  theory  in  which  chemical  action  is  concerned. 
121 


MKMOIRS    OX 

ply  Carnot's  unmodified  investigation  of  the  relation  between 
the  mechanical  effect  produced  and  the  thermal  cireiuustamvs 
from  which  it  originates,  in  the  case  of  an  expansive  enirine 
working  within  an  infinitely  small  range  of  temperatures.  The 
simplest  investigation  of  the  consequences  of  the  first  proposi- 
tion in  this  application,  which  has  occurred  to  me,  is  the  fol- 
lowing, being  merely  the  modification  of  an  analytical  expres- 
sion of  Carnot's  axiom  regarding  the  permanence  of  heat,  which 
was  given  in  my  former  paper,*  required  to  make  it  express, 
not  Carnot's  axiom,  but  Joule's. 

20.  Let  us  suppose  a  massf  of  any  substance,  occupying  a 
volume  v,  under  a  pressure  p  uniform  in  all  directions,  and  at 
a  temperature  /,  to  expand  in  volume  to  v  +  <lr.  and  to  rise  in 
temperature  to  t  +  dt.  The  quantity  o*f  work  which  it  will  pro- 
duce will  be  pfo  . 

and  the  quantity  of  heat  which  must  be  added  to  it  to  make  its 
temperature  rise  during  the  expansion  to  t  +  dt  may  be  de- 
noted by  Mdv  +  Sdt. 

The  mechanical  equivalent  of  this  is 

./<.)/'//•  4-  -V'//). 

if  J  denote  the  mechanical  equivalent  of  a  unit  of  heat.  Hence 
the  mechanical  measure  of  the  total  external  effect  produced  in 
the  circumstances  is 

(p-JM)dv-J.\<lt. 

The  total  external  effect,  after  any  finite  amount  of  expansion, 
accompanied  by  any  continuous  change  of  temperature,  has 
taken  place,  will  consequently  be,  in  mechanical  ti-rm-. 


where  we  must  suppose  /  to  vary  with  v,  so  as  to  be  (lie  actual 
temperature  of  the  medium  at  each  instant,  ami  the  integration 
with  reference  to  v  must  be  performed  between  limits  fin-re- 
sponding to  the  initial  and  final  volumes.  Now  if,  at  any  sub- 
sequent time,  the  volume  and  temperature  of  the  medium  be- 
come what  they  were  at  the  beginning,  however  arbitrarily 

•  "  Account  of  Carnot's  Theory."  foot-note  on  §  26. 

f  This  may  have  part*  consisting  of  different  substances,  or  of  the  same 
substance  in  different  states,  provided  the  temperature  of  all  be  the  same. 
See  below,  pan  iii.,  g  58-56. 

m 


THE    SECOND    LAW    OF    THERMODYNAMICS 

they  may  have  been  made  to  vary  in  the  period,  the  total  ex- 
ternal effect  must,  according  to  Prop.  I.,  amount  to  nothing  ; 
and  hence  (p-JM}dv-JNdt 

must  be  the  differential  of  a  function  of  two  independent  varia- 
bles, or  we  must  have 

d(p-JM)    d(-JN)  m 

dt  dv 

this  being  merely  the  analytical  expression  of  the  condition,  that 
the  preceding  integral  may  vanish  in  every  case  in  which  the 
initial  and  final  values  of  v  and  t  are  the  same,  respectively. 
Observing  that  J  is  an  absolute  constant,  we  may  put  the  result 
into  the  form 

dl-  Tl**£    dN\  (2) 

dt  ~     \  dt       dv  ) 

This  equation  expresses,  in  a  perfectly  comprehensive  manner, 
the  application  of  the  first  fundamental  proposition  to  the  ther- 
mal and  mechanical  circumstances  of  any  substance  whatever, 
under  uniform  pressure  in  all  directions,  when  subjected  to 
any  possible  variations  of  temperature,  volume,  and  pressure. 

21.  The  corresponding  application  of  the  second  fundamental 
proposition  is  completely  expressed  by  the  equation 

!=**  <") 

where  n  denotes  what  is  called  "  Carnot's  function,"  a  quantity 
which  has  an  absolute  value,  the  same  for  all  substances  for 
any  given  temperature,  but  which  may  vary  with  the  temper- 
ature in  a  manner  that  can  only  be  determined  by  experiment. 
To  prove  this  proposition,  it  may  be  remarked  in  the  first  place 
that  Prop.  II.  could  not  be  true  for  every  case  in  which  the 
temperature  of  the  refrigerator  differs  infinitely  little  from  that 
of  the  source,  without  being  true  universally.  Now,  if  a  sub- 
stance be  allowed  first  to  expand  from  v  to  v  +  dv,  its  temper- 
ature being  kept  constantly  t ;  if,  secondly,  it  be  allowed  to 
expand  further,  without  either  emitting  or  absorbing  heat 
till  its  temperature  goes  down  through  an  infinitely  small 
range,  to  t—  r;  if,  thirdly,  it  be  compressed  at  the  constant 
temperature  t  —  T,  so  much  (actually  by  an  amount  differing 
from  dv  by  only  an  infinitely  small  quantity  of  the  second  or- 
der), that  when,  fourthly,  the  volume  is  further  diminished  to 
123 


MKMOIRS    <>N 

r  without  the  medium's  being  allowed  to  either  emit  or  absorb 
heat.  its  temperature  may  be  exactly  /:  it  may  ho  considered 
as  constituting  a  thermo-dynamic  engine  which  fulfils  Carnot's 
condition  of  complete  reversibility.  Hence,  by  Prop.  II.,  it 
must  produce  the  same  amount  of  work  for  the  same  quantity 
of  heat  absorbed  in  the  first  operation,  as  any  other  substam  ••• 
similarly  operated  upon  through  the  same  range  of  temper- 

atures.    But   •    T.di'  is  obviously  the  whole  work  done  in  tin- 

complete  cycle,  and  (by  the  definition  of  M  in  $  20)  Mdv  is  the 
quantity  of  heat  absorbed  in  the  first  operation.  Hence  the 

value  of  ,//,  dp 

dtr'dv       ft 


must  be  the  same  for  all  substances,  with  the  same  values  of  / 
and  r;  or,  since  r  is  not  involved  except  as  a  factor,  we  must  have 

4 

<//  ... 

-="' 

where  p  depends  only  on  /;  from  which  we  conclude  the  prop- 
osition which  was  to  be  proved.  i 

22.  The  very  remarkable  theorem  that  "jT  must  be  the  same 

for  all  substances  at  the  same  temperature  was  first  given 
(although  not  in  precisely  the  same  terms)  by  Carnot,  and  de- 
monstrated by  him,  according  to  the  principles  he  adopted. 
We  have  now  seen  that  its  truth  may  be  satisfactorily  cstal.- 
lished  without  adopting  the  false  part  of  his  principles.  Hence 
all  Carnot's  conclusions,  and  all  conclusions  derived  by  others 
from  his  theory,  which  depend  merely  on  equation  (';$).  require 
no  modification  when  the  dynamical  theory  is  adopted.  Tims. 
all  the  conclusions  contained  in  Sections  I..  11.  .and  III.  of 
the  Appendix  to  my  "Account  of  Carnot's  Theory."  and  in  the 
paper  immediately  following  it  in  the  YV/n/>v//7/W*.  entitled 
"Theoretical  Considerations  on  the  Effect  of  Pressure  in  Lower- 
ing the  Kreexing-point  of  Water,"  by  my  elder  brother,  still  hold. 
Also,  we  sec  that  Carnot's  expressii-n  for  the  mechanienl  eiT.-.-t 
derivable  from  a  given  quantity  of  heat  l»v  means  of  a  perfect 
124 


THE    SECOND    LAW    OF    THERMODYNAMICS 

engine  in  which  the  range  of  temperatures  is  infinitely  small, 
expresses  truly  the  greatest  effect  which  can  possibly  be  ob- 
tained in  the  circumstances  ;  although  it  is  in  reality  only  an 
infinitely  small  fraction  of  the  whole  mechanical  equivalent  of 
the  heat  supplied ;  the  remainder  being  irrecoverably  lost  to 
man,  and  therefore  "  wasted,"  although  not  annihilated. 

23.  On  the  other  hand,  the  expression  for  the  mechanical 
effect  obtainable  from  a  given  quantity  of  he«t  entering  an  en- 
gine from  a  "source"  at  a  given  temperature,  when  the  range 
down  to  the  temperature  of  the  cold  part  of  the  engine  or  the 
"refrigerator"  is  finite,  will  differ  most  materially  from  that 
of  Carnot ;  since,  a  finite  quantity  of  mechanical  effect  being 
now  obtained  from  a  finite  quantity  of  heat  entering  the  engine, 
a  finite  fraction  of  this  quantity  must  be  converted  from  heat 
into  mechanical  effect.  The  investigation  of  this  expression, 
with  numerical  determinations  founded  on  the  numbers  de- 
duced from  Regnault's  observations  on  steam,  which  are  shown 
in  Tables  I.  and  II.  of  my  former  paper,  constitutes  the  second 
part  of  the  paper  at  present  communicated. 


PART  II 

On  the  Motive  Power  of  Heat  through  Finite  Ranges  of 
Temperature 

24.  It  is  required  to  determine  the  quantity  of  work  which  a 
perfect  engine,  supplied  from  a  source  at  any  temperature,  8, 
and  parting  with  its  waste  heat  to  a  refrigerator  at  any  lower 
temperature,  T,  will  produce  from  a  given  quantity,  //,  of  heat 
drawn  from  the  source. 

25.  We  may  suppose  the  engine  to  consist  of  an  infinite  num- 
ber of  perfect  engines,  each  working  within  an  infinitely  small 
range  of  temperature,  and  arranged  in  a  series  of  which  the 
source  of  the  first  is  the  given  source,  the  refrigerator  of  the 
last  the  given  refrigerator,  and  the  refrigerator  of  each  inter- 
mediate engine  is  the  source  of  that  which  follows  it  in  the 
series.     Each  of  these  engines  will,  in  any  time,  emit  just  as 
much  less  heat  to  its  refrigerator  than  is  supplied  to  it  from  its 
source,  as  is  the  equivalent  of  the  mechanical  work  which  it 
produces.     Hence  if  t  and  t  +  dt  denote  respectively  the  tem- 
peratures of  the  refrigerator  and  source  of  one  of  the  inter- 

125 


MK  M(>  IRS    ON 

mediate  engines,  and  if  q  denote  the  quantity  of  heat  which 
this  engine  discharges  into  its  refrigerator  in  any  time.  ami 
q  +  dq  the  quantity  which  it  draws  from  its  source  in  the  same 
time,  the  quantity  of  work  which  it  produces  in  that  time  will 
be  Jdq  according  to  Prop.  I.,  and  it  will  also  be  qpdt  according 
to  the  expression  of  Prop.  II.,  investigated  in  §  •„'!  :  and  there- 
fore we  must  have 


Hence,  supposing  that  the  quantity  of  heat  supplied  from  the 
first  source,  in  the  time  considered  is  //.  we  timl  hy  integration 


But  the  value  of  q,  when  /=  T,  is  the  final  remainder  dis- 
charged into  the  refrigerator  at  the  temperature  T;  and  there- 
fore, if  this  be  denoted  by  R,  we  have 


from  which  we  deduce 

R  =  m-jf'l>"«  (r.) 

Now  the  whole  amount  of  work  produced  will  be  the  mechani- 
cal equivalent  of  the  quantity  of  heat  lost  ;  and,  therefore,  if 
this  be  denoted  by  II  .  we  have 

\\'  =  J(/I-R),  (7) 

and  consequently,  by  (C), 

ir  =  J//{l-f-i/J^//  (• 
26.  To  compare  this  with  the  expression  II  J  f«lt.  for  the 

duty  indicated  by  Carnot's  theory,*  we  may  expand  the  e\po. 
nential  in  the  preceding  equation,  by  the  usual  series.  We  thus 


find 

M 


I  • 

where  6  =  ,  / 

JT 


v 

, 

J 

This  shows  that  the  work  really  produced,  which  always  falls 
short  of  the  duty  indicated  by  Carnot's  theory,  approaches 


*  "Account,"  etc..  Equation  7,  £  HI. 
126 


THE    SECOND    LA^y    OF    THERMODYNAMICS 

more  and  more  nearly  to  it  as  the  range  is  diminished  ;  and  ul- 
timately, when  the  range  is  infinitely  small,  is  the  same  as  jf 
Carnot's  theory  required  no  modification,  which  agrees  with 
the  conclusion  stated  above  in  §  22. 

27.  Again,  equation  (8)  shows  that  the  real  duty  of  a  given 
quantity  of  heat  supplied  from  the  source  increases  with  every 
increase  of  the  range  ;  but  that  instead  of  increasing  indefinitely 

in  proportion  to  /  \idt,  as  Carnot's  theory  makes  it  do,  it  never 
reaches  the  value  JH,  but  approximates  to  this  limit,  as  /  *  pelt 

is  increased  without  limit.  Hence  Carnot's  remark*  regarding 
the  practical  advantage  that  may  be  anticipated  from  the  use 
of  the  air-engine,  or  from  any  method  by  which  the  range  of 
temperatures  may  be  increased,  loses  only  a  part  of  its  impor- 
tance, while  a  much  more  satisfactory  view  than  his  of  the  prac- 
tical problem  is  afforded.  Thus  we  see  that,  although  the 
full  equivalent  of  mechanical  effect  cannot  be  obtained  even  by 
means  of  a  perfect  engine,  yet  when  the  actual  source  of  heat 
is  at  a  high  enough  temperature  above  the  surrounding  objects, 
we  may  get  more  and  more  nearly  the  whole  of  the  admitted 
heat  converted  into  mechanical  effect,  by  simply  increasing  the 
effective  range  of  temperature  in  the  engine. 

28.  The  preceding  investigation  (§  25)  shows  that  the  value 
of  Carnot's  function,  /u,  for  all  temperatures  within  the  range 
of  the  engine,  and  the  absolute  value  of  Joule's  equivalent,  J, 
are  enough  of  data  to  calculate  the  amount  of  mechanical  effect 
of  a  perfect  engine  of  any  kind,  whether  a  steam-engine,  an  air- 
engine,  or  even  a  tbermo  -  electric  engine  ;  since,  according  to 
the  axiom  stated  in  §  12,  and  the  demonstration  of  Prop.  II., 
no  inanimate  material  agency  could  produce  more  mechanical 
effect  from  a  given  quantity  of  heat,  with  a  given  available 
range  of  temperatures,  than  an  engine  satisfying  the  criterion 
stated  in  the  enunciation  of  the  proposition. 

29.  The  mechanical  equivalent  of  a  thermal  unit  Fahrenheit, 
or  the  quantity  of  heat  necessary  to  raise  the  temperature  of  a 
pound  of  water  from  32°  to  33°  Fahr.,  has  been  determined  by 
Joule  in  foot-pounds  at  Manchester,  and  the  value  which  he 
gives  as  his  best  determination  is  772.69.     Mr.  Rankine  takes, 

*  "Account,"  etc.     Appendix,  Section  iv. 
127 


MEMOIRS    ON 

as  the  result  of  Joule's  determination,  77--J,  which  he  estimates 
must  be  within  yfa  of  its  own  amount,  of  tin-  truth.  If  \\c 
take  ??2f  as  the  number,  we  find,  by  multiplying  it  by  |,  i:>'.">  as 
the  equivalent  of  the  thermal  unit  Centigrade,  which  is  taken 
as  the  value  of  J  in  the  numerical  applications  contained  in  tin- 
present  paper. 

30.  With  regard  to  the  determination  of  the  values  of  /<  for 
different  temperatures,  it  is  to  be  remarkr»l  that  equation  (4) 
shows  that  this  might  be  done  by  experiments  upon  any  sub- 
stance whatever  of  indestructible  texture,  and  indicates  exactly 
the  experimental  data  required  in  each  case.  For  instance,  l>v 
first  supposing  the  medium  to  be  air  ;  and  again,  by  supposing 
it  to  consist  partly  of  liquid  water  and  partly  of  saturated  vapor. 
we  deduce,  as  is  shown  in  Part  III.  of  this  paper,  the  tw<<  •  \- 
pressions  (G),  given  in  §  30  of  my  former  paper  ("  Account  of 
('arnot's  Theory"),  for  the  value  of  p.  at  any  temperature.  A< 
yet  no  experiments  have  been  made  upon  air  which  afford  the 
required  data  for  calculating  the  value  of  ^  through  any  exten- 
sive range  of  temperature;  but  for  temperatures  between  .~>n 
and  60°  Fahr.,  Joule's  experiments*  on  the  heat  evolved  1>\  tin- 
expenditure  of  a  given  amount  of  work  on  the  compression  of 
air  kept  at  a  constant  temperature,  afford  the  most  direct  data 
for  this  object  which  have  yet  been  obtained  ;  since,  if  Q  be  the 
quantity  of  heat  evolved  by  the  compression  of  a  fluid  subject 
to  "the  gaseous  laws"  of  expansion  and  compressibility,  IT  the 
amount  of  mechanical  work  spent,  and  /  the  constant  temper- 
ature of  the  fluid,  we  have  by  (11)  of  §  49  of  my  former  paper. 

n.  /•: 


which  is  in  reality  a  simple  consequence  of  the  other  expression 
for  ft  in  terms  of  data  with  reference  to  air.  Remark!  upon 
the  determination  of  fi  by  such  experiments,  and  by  another 
class  of  experiments  on  air  originated  liy  Joule,  are  reserved 
for  a  separate  communication,  which  I  hope  to  be  able  to  make 
to  the  Royal  .Society  on  another  occasion. 

31.  The  second  of  the  expressions  (6),  in  §  30  of  my  former 
paper,  or  the  equivalent  expression  (32),  given  below  in  the 

*  "On  the  Change*  of  Temperature  produced  by  Hn-  Rarefaction  and 
Condensation  of  Air,"  Phil.  Mug.,  vol.  xxvi  ,  May,  1845. 

in 


THE    SECOND    LAW    OF    THERMODYNAMICS 

present  paper,  shows  that  /*  may  be  determined  for  any  tem- 
perature from  determinations  for  that  temperature  of — 

(1)  The  rate  of  variation  with  the  temperature,  of  the  press- 
ure of  saturated  steam. 

(2)  The  latent  heat  of  a  given  weight  of  saturated  steam. 

(3)  The  volume  of  a  given  weight  of  saturated  steam. 

(4)  The  volume  of  a  given  weight  of  water. 

The  last  mentioned  of  these  elements  may,  on  account  of  the 
manner  in  which  it  enters  the  formula,  be  taken  as  constant, 
without  producing  any  appreciable  effect  on  the  probable  accu- 
racy of  the  result. 

32.  Regnault's  observations  have  supplied  the  first  of  the 
data  with  very  great  accuracy  for  all  temperatures  between 
—  32°  Cent,  and  230°. 

33.  As  regards  the  second  of  the  data,  it  must  be  remarked 
that  all  experimenters,  from  Watt,  who  first  made  experiments 
on  the  subject,  to  Regnault,  whose  determinations  are  the  most 
accurate  and  extensive  that  have  yet  been  made,  appear  to  have 
either  explicitly  or  tacitly  assumed  the  same  principle  as  that  of 
Carnot  which  is  overturned  by  the  dynamical  theory  of  heat ; 
inasmuch  as  they  have  defined  the  "  total  heat  of  steam"  as  the 
quantity  of  heat  required  to  convert  a  unit  of  weight  of  water 
at  0°  into  steam  in  the  particular  state  considered.    Thus  Reg- 
nault, setting  out  with  this  definition  for  "the  total  heat  of 
saturated  steam,"  gives  experimental  determinations  of  it  for 
the  entire  range  of  temperatures  from  0°  to  230°  ;  and  he  de- 
duces the  "  latent  heat  of  saturated  steam  "  at  any  temperature, 
from  the  "total  heat,"  so  determined,  by  subtracting  from  it 
the  quantity  of  heat  necessary  to  raise  the  liquid  to  that  tem- 
perature.    Now,  according  to  the  dynamical  theory,  the  quan- 
tity of  heat  expressed  by  the  preceding  definition  depends  on 
the  manner  (which  may  be  infinitely  varied)  in  which  the  speci- 
fied change  of  state  is  effected  ;  differing  in  different  cases  by 
the  thermal  equivalents  of  the  differences  of  the  external  me- 
chanical effect  produced  in  the  expansion.     For  instance,  the 
total  quantity  of  heat  required  to  evaporate  a  quantity  of  water 
at  0°,  and  then,  keeping  it  always  in  the  state  of  saturated  va- 
por,* bring  it  to  the  temperature  100°,  cannot  be  so  much  as 

*  See  below  (Part  III.,  §  58),  where  the  "negative"  specific  heat  of  sat- 
urated steam  is  investigated.     If  the  mean  value  of  this  quantity  between 
0°  and  100°  were  -1.5  (and  it  cannot  differ  much  from  this)  there  would 
i  129 


MEMOIRS    ON 

three-fourths  of  the  quantity  required,  first,  to  raise  the  tem- 
perature of  the  liquid  to  loo  ,  ami  then  evaporate  it  at  that 
temperature  ,  and  yet  either  quantity  is  expressed  by  what  is 
generally  received  as  a  definition  of  the  " total  heat"  of  the 
saturated  vapor.  To  find  what  it  is  that  is  really  determined 
as  "total  heat"  of  saturated  steam  in  lleirnault's  researches,  it 
is  only  necessary  to  remark,  that  the  measurement  actually 
made  is  of  the  quantity  of  heat  emitted  by  a  certain  weight  of 
water  in  passing  through  a  calorimetrical  apparatus,  which  it 
enters  as  saturated  steam,  and  leaves  in  the  liquid  state,  the 
result  being  reduced  to  what  would  have  been  found  if  the  final 
temperature  of  the  water  had  been  exactly  0°.  For  there  bein^ 
no  external  mechanical  effect  produced  (other  than  that  of 
sound,  which  it  is  to  be  presumed  is  quite  inappreciable),  the 
only  external  effect  is  the  emission  of  heat.  This  must,  there- 
fore, according  to  the  fundamental  proposition  of  the  dynam- 
ical theory,  be  independent  of  the  intermediate  agencies.  It 
follows  that,  however  the  steam  may  rush  through  the  calo- 
rimeter, and  at  whatever  reduced  pressure  it  may  actually  he 
condensed,*  the  heat  emitted  externally  must  be  exactly  t he 

be  150  units  of  heat  emitted  by  a  pound  of  saturated  vapor  in  having  its 
temperature  raised  (by  compression)  from  0°  t<>  1(M»  .     Tin-  latent  ln-at  <>l 
the  vapor  at  0°  being  606.5.  the  tiiml  quantity  of  heat  required  to  OOB1 
pound  of  water  at  0°  into  saturated  steam  at  100  ,  in  tin*  first  <>f  tli> 
mentioned  in  the  text,  would  consequent])'  he  456.5.  which  is  only  about  J 
of  the  quantity  687  found  us  "  the  total  heat"  of  the  saturated  vapor  at 
100°,  by  Regnault. 

*  If  the  steam  have  to  rush  through  a  long  flne  tulic,  or  through  a  small 
aperture  within  the  calorimctrical  apparatus,  it-  pressure  will  In  dimin- 
ished l>efore  it  is  condensed  ;  and  there  will,  therefore,  in  two  parts  <>f  ihe 
calorimeter  be  satuntrd  Meani  at  different  trmperatuic*  (a«.  for  invfirce. 
would  be  the  case  if  steam  from  a  high  pressure  holler  were  distilled  into 
the  open  air);  yet.  on  account  of  the  heat  developed  l>y  the  tlui<l  friction, 
winch  would  he  precisely  the  equivalent  »f  tin-  inerhan:ca;  elTect  of  the 
expansion  wasted  in  tin;  rushinp.  the  heat  monsnn-d  l.y  the  calnrim.  t.  r 
would  he  precisely  the  same  MS  if  the  condensation  took  place  at  a  pressure 
not  appreciably  lower  than  that  of  ihe  entering  steam  The  ciicnm-' 
of  such  a  case  have  l*cn  overlooked  by  Clausiu«  (IWirendorlTs  Annul,  n. 
1850.  No.  4.  p.  510).  when  lie  expresses  with  some  d<>ul>l  the  opinion  that 
ill.-  latent  heat  of  saturated  steam  will  he  truly  found  fr-m  K«  ^naull's 
"total  heat."  hy  deducting  "  the  sensible  heat ;"  ami  give*  as  i\  reason  that, 
in  the  actual  experiments,  the  condensation  must  have  token  place  "under 
the  same  pressure,  or  nearly  under  the  same  pressure."  as  the  evaporation 
The  question  is  not,  Did  the  condensation  take  place  at  a  lover  prettui-f  /',.n, 
MO 


THE    SECOND    LAW    OF    THERMODYNAMICS 

same  as  if  the  condensation  took  place  under  the  full  pressure 
of  the  entering  saturated  steam  ;  and  we  conclude  that  the 
total  heat,  as  actually  determined  from  his  experiments  by  Reg- 
nault,  is  the  quantity  of  heat  that  would  be  required,  first  to 
raise  the  liquid  to  the  specified  temperature,  and  then  to  evap- 
orate it  at  that  temperature  ;  and  that  the  principle  on  which 
he  determines  the  latent  heat  is  correct.  Hence,  through  the 
range  of  his  experiments — that  is,  from  0°  to  230° — we  may  con- 
sider the  second  of  the  data  required  for  the  calculation  of  /j. 
as  being  supplied  in  a  complete  and  satisfactory  manner. 

34.  There  remains  only  the  third  of  the  data,  or  the  volume 
of  a  given  weight  of  saturated  steam,  for  which  accurate  exper- 
iments through  an  extensive  range  are  wanting ;  and  no  ex- 
perimental researches  bearing  on  the  subject  having  been  made 
since  the  time  when  my  former  paper  was  written,  I  see  no 
reason  for  supposing  that  the  values  of  p  which  I  then  gave  are 
not  the  most  probable  that  can  be  obtained  in  the  present  state 
of  science  ;  and,  on  the  understanding  stated  in  §  33  of  that 
paper,  that  accurate  experimental  determinations  of  the  den- 
sities of  saturated  steam  at  different  temperatures  may  indicate 
considerable  errors  in  the  densities  which  have  been  assumed 
according  to  the  "gaseous  laws," and  may  consequently  render 
considerable  alterations  in  my  results  necessary,  I  shall  still  con- 
tinue to  use  Table  I.  of  that  paper,  which  shows  the  values  of 
H  for  the  temperatures  £,  H,  £•£... 230$,  or,  the  mean  values 
of  p  for  each  of  the  230  successive  Centigrade  degrees  of  the 
air-thermometer  above  the  freezing-point,  as  the  basis  of  nu- 
merical applications  of  the  theory.  It  may  be  added,  that  any 
experimental  researches  sufficiently  trustworthy  in  point  of  ac- 
curacy, yet  to  be  made,  either  on  air  or  any  other  substance, 
which  may  lead  to  values  of  p.  differing  from  those,  must  be 
admitted  as  proving  a  discrepancy  between  the  true  densities 
of  saturated  steam,  and  those  which  have  been  assumed.* 

that  of  the  entering  steam  ?  but.  Did  Regnnult  maketJie  steam  work  an  engine 
in  parsing  through  the  calorimeter,  or  was  there  so  much  noise  of  steam  rush- 
ing through  it  as  to  convert  an  appreciable  portion  of  tJie  total  heat  into  ex- 
ternal mechanical  effect?  And  a  negative  answer  to  this  is  a  sufficient  reason 
for  adopting  with  certainty  the  opinion  that  the  principle  of  his  determina- 
tion of  the  fatent  heat  is  correct. 

*  I  cannot  see  that  any  hypothesis,  such  as  that  ndopted  by  Clausius 
fundamentally  in  his  investigations  on  this  subject,  and  leading,  as  he  shows, 
to  determinations  of  the  densities  of  saturated  steam  at  different  temper- 
131 


MEMOIRS    ON 

35.  Table  II.  of  my  former  paper,  which  shows  the  values  of 
f  pit  for  t  =  l,  t  =  2,  /  =  3,  ...  /  =  231,  renders  the  calcula- 

tion of  the  mechanical  effect  derivable  from  a  given  quantity 
of  heat  by  means  of  a  perfect  engine,  with  any  given  range  in- 
cluded between  the  limits  0  and  231,  extremely  easy  :  since  the 
quantity  to  be  divided  by  J*  in  the  index  of  the  exponential  in 
the  expression  (8)  will  be  found  by  subtracting  the  number  in 
that  table  corresponding  to  the  value  of  T,  from  that  corre- 
sponding to  the  value  of  S. 

36.  The  following  tables  show  some  numerical  results  which 
have  been  obtained  in  this  way,  with  a  few  (contained  in  the 

lower  part  of  the  second  table)  calculated  from  values  of  f  n<lt 

estimated  for  temperatures  above  230°,  roughly,  according  to 
the  rate  of  variation  of  that  function  within  the  experimental 
limits. 

37.  Explanation  of  the  Tables. 

Column  I.  in  each  table  shows  the  assumed  ranges. 
Column  II.  shows  ranges  deduced  by  means  of  Table  II.  of 

the  former  paper,  so  that  the  value  otf'ndt  for  each  may  be 

the  same  as  for  the  corresponding  range  shown  in  column  I. 

Column  III.  shows  what  would  be  the  duty  of  a  unit  of  heat 
if  Carnot's  theory  required  no  modification  (or  the  actual  duty 
of  a  unit  of  heat  with  additions  through  the  ran^e.  to  compen- 
sate forAhe  quantities  converted  into  mechanical  effect). 


which  indicate  enormous  deviations  from  the  gaseous  laws  of  varia- 
tion with  temperature  and  pressure,  is  more  probable,  or  is  probably  nearer 
the  truth,  than  that  the  density  of  saturated  steam  does  follow  these  laws 
as  it  is  usually  assumed  to  do.  In  the  present  state  of  science  it  would 
perhaps  be  wrong  to  say  that  either  hypothesis  is  more  probable  than  tin- 
other  [or  that  the  rigorous  truth  of  cither  hypothesis  is  probable  at  all]. 

*  It  ought  to  be  remarked,  that  as  the  unit  of  force  implied  in  the  deter- 
minations of  ft  is  the  weight  of  a  pound  <>f  mutter  at  Paris,  and  the  unit  <>f 
force  in  terms  of  which  ./  is  expressed  Is  the  wcipht  of  a  pound  at  Man- 
chester. these  numbers  ought  in  strictness  to  be  modified  so  as  to  express 
the  values  in  terms  of  a  common  unit  of  force  ;  but  as  the  force  of  gravity 
at  Parii  differs  by  less  than  ^j  of  its  own  value  from  the  force  of  gravity 
at  Manchester,  this  correction  will  be  much  less  Minn  the  probable  errors 
from  other  sources,  and  may  therefore  be  neglected. 


THE    SECOND    LAW    OF    THERMODYNAMICS 


Column  IV.  shows  the  true  duty  of  a  unit  of  heat,  and  a  com- 
parison of  the  numbers  in  it  with  the  corresponding  numbers 
in  Column  III.  shows  how  much  the  true  duty  falls  short  of 
Carnot's  theoretical  duty  in  each  case. 

Column  VI.  is  calculated  by  the  formula 


=  t~  1390/2'  /'*#> 


the  successive  values  shown 


where  e  =  2.  71828,  and  for  / 
v 
in  Column  III.  are  used. 

Column  IV.  is  calculated  by  the  formula 

W=  1390  (l-R) 
from  the  values  of  1  —  R  shown  in  Column  V. 

38.   Table  of  the  Motive  Power  of  Heat. 


Range  of  Temperatures 

III 

Duty  of  a  unit 
of  beat 
through  the 
whole  range 

IV 

Duty  of  a  unit 
of  heat  sup- 
plied  from 
the  source 

V 
Quantity  of  heat 
converted  into 
mechanical 
effect 

VI 

Quantity  of 
heat  wasted 

I 

II 

S 

T 

S 

r 

fS 

IP 

l-R 

R 

0 

a 

0 

0 

ft.  Ibs. 

ft.  -Ibs. 

1 

0 

31.08 

30 

4.960 

4.948 

.003.56 

.99644 

10 

0 

40.86 

30 

48.987 

48.1 

.0346 

.9654 

20 

0 

51.7 

30 

96656 

93.4 

.067 

.933 

30 

(1 

62.6 

30 

143.06 

136 

.098 

.902 

40 

0 

73.6 

30 

188.22 

176 

.127 

.873 

50 

0 

84.5 

30 

232.18 

214 

.154 

.846 

60 

0 

954 

30 

27497 

249 

.179 

.821 

70 

0 

106.3 

30 

316.64 

283 

.204 

.796 

80 

0 

117.2 

BO 

357.27 

315 

.227 

.773 

90 

0 

128.0 

30 

396.93 

345 

.248 

.752 

100 

0 

138.8 

30 

435.69 

374 

.269 

.731 

110 

0 

149.1 

30 

473.62 

401 

.289 

.711 

120 

0 

160.3 

30 

510.77 

427 

.308 

.692 

130 

0 

171.0 

80 

547.21 

452 

.325 

.675 

140 

0 

181.7 

BO 

582.98 

476 

.343 

.657 

150 

0 

192.3 

30 

618.14 

499 

.359 

.641 

160 

0 

2030 

30 

652.74 

521 

.375 

.625 

170 

0 

213.6 

30 

686.80 

542 

.390 

.610 

180 

Q 

224.2 

3(1 

720.39 

562 

.404 

.596 

190 

0 

190 

753.50 

582 

.418 

.582    ' 

200 

0 

200 

(1 

786.17 

600 

.432 

.568 

210 

0 

210 

0 

818.45 

619 

.445 

.555 

220 

0 

220 

0 

850.34 

636 

.457 

.542 

230 

(1 

230 

0 

88187 

653 

.470 

.530 

MKMnlKS    ON 


39.  Supplementary  Table  of  the  Motive  Power  of  11  ///. 


III 

IT 

V 

VI 

Range  or  Temperatures 

Duty  of  a  unit     IMHV  of  •  unit 
of  beat             or  beat  sup- 

Quantity  of  lie*l 
convened  Into 

Quantity  of 

I 

II 

whole  range 

the  source 

effect 

L-  M!    \\    i.-tr-: 

0 

/ 

s 

T 

£** 

W 

\-R 

R 

0 

0 

- 

ft  Ita. 

ft  It*. 

101.1 

0 

140 

M 

»::;•  11 

877 

.271 

105.8 

II 

280 

LOO 

446.2 

882 

.275 

800 

0 

800 

(i 

1099 

757 

.64B 

.455 

400 

0 

400 

n 

1  :!'.>:, 

879 

.682 

M 

500 

0 

500 

n 

1690 

979 

.704 

M 

600 

0 

600 

0 

1980 

LOW 

.762 

888 

00 

0 

00 

(1 

30 

1890 

1.000 

.000 

40.  Taking  the  range  30°  to  140°  as  an  example  suitable  to 
the  circumstances  of  some  of  the  best  steam-engines  that  have 
yet  been  made  (see  Appendix  to  "  Account  of  Carnot's  Theory," 
sec.  v.),  we  find  in  Column  III.,  of  the  supplementary  table,  377 
ft. -11)8.  as  the  corresponding  duty  of  a  unit  of  heat  instead  of 
440,  shown  in  Column  III.,  which  is  Carnot's  theoretical  duty. 
We  conclude  that  the  recorded  performance  of  the  Fowey-Con- 
sols  engine  in  1845,  instead  of  being  only  ."»?i  per  cent,  amounted 
n-ally  to  67  per  cent.,  or  \  of  the  duty  of  a  perfect  engine  with 
the  same   range  of  temperature;    and  this  duty  being  .Mil 
(rather  more  than  J )  of  the  whole  equivalent  of  the  heat  used  ; 
we  conclude  further,  that  ^  or  18  per  cent,  of  the  whole  lu-at 
supplied  was  actually  converted  into  mechanical  effort  In  that 
steam-engine. 

41.  The  numbers  in  the  lower  part  of  the  supplementary 
table  show  the  great  advantage  that  may  be  anticipated  from 
the  perfecting  of  the  air-engine,  or  any  other  kind  of  thermo- 
•iynamic  engine  in  which  the  range  of  the  temperature  can  be 
increased  much  beyond  the  limits  actually  attainable  in  steam- 
engines.     Thus  an  air-engine,  with  its  hot  part  at  600°,  and  it- 
cold  part  at  0°  Cent.,  working  with  perfect  economy,  would 
convert  76  per  cent,  of  the  whole  heat  used  into  median i( -al  t  f- 

or  working  with  such  economy  as  has  been  estimated  for 
the  Fowey-Consols  engine — that  is.  producing  <>7  per  cent,  of 
the  theoretical  duty  corresponding  to  its  range  of  temperature — 
LM 


THE    SECOND    LAW    OF    THERMODYNAMICS 

would  convert  51  per  cent,  of  all  the  heat  used  into  mechanical 
effect. 

42.  It  was  suggested  to  me  by  Mr.  Joule,  in  a  letter  dated 
December  9,  1848,  that  the  true  value  of  p.  might  be  "inversely 
as  the  temperatures  from  zero  ;"  *  and  values  for  various  tem- 
peratures calculated  by  means  of  the  formula, 


were  given  for  comparison  with  those  which  I  had  calculated 
from  data  regarding  steam.  This  formula  is  also  adopted  by 
Clausius,  who  uses  it  fundamentally  in  his  mathematical  inves- 
tigations. If  p.  were  correctly  expressed  by  it,  we  should  have 


and  therefore  equations  (1)  and  (2)  would  become 

ir  =  j£^£,  (12) 


43.  The  reasons  upon  which  Mr.  Joule's  opinion  is  founded, 
that  the  preceding  equation  (11)  may  be  the  correct  expression 

TJT 

*  If  we  take  /*=A;T— — -  where  k  may  be  any  constant,  we  find 

1  "T~  J^t 

k 
— 
—  J  I 


'(&' 


which  is  the  formula  I  gave  when  this  paper  was  communicated.  I  have 
since  remarked  that  Mr.  Joule's  hypothesis  implies  essentially  that  the  co- 
efficient k  must  be  as  it  is  taken  in  the  text,  the  mechanical  equivalent  of  a 
thermal  unit.  Mr.  Rankine,  in  a  letter  dated  March  27,  1851,  informs  me 
that  he  has  deduced,  from  the  principles  laid  down  in  his  paper  communi- 
cated last  year  to  this  Society,  an  approximate  formula  for  the  ratio  of  the 
maximum  quantity  of  heat  converted  into  mechanical  effect  to  the  whole 
quantity  expended,  in  an  expansive  engine  of  any  substance,  which,  on 
comparison,  I  find  agrees  exactly  with  the  expression  (12)  given  in  the 
text  as  a  consequence  of  the  hypothesis  suggested  by  Mr.  Joule  regarding 
the  value  of  p  at  any  temperature. — [April  4,  1851.] 
135 


MEMOIRS    ON 

for  Carnot's  function,  although  the  values  calculated  by  means 
of  it  differ  considerably  from  those  shown  in  Table  I.  of  my 
former  paper,  form  the  subject  of  a  communication  which  I 
hope  to  have  an  opportunity  of  laying  before  the  Royal  Society 
previously  to  the  close  of  the  present  session. 


PART  III. 

Applications  of  lite  Dynamical  '/'//• ///•//  fn  >*/n/tli.«/i  Hrtations  be- 
lii'fi'ii  the  ritysicul  J'n>/»rfi>s  of  all  >W/*A- 

44.  The  two  fundamental  equations  of  the  dynamical  theory 
of  heat,  investigated  above,  express  relations  between  quanti- 
ties of  heat  required  to  produce  changes  of  volume  and  tem- 
perature in  any  material  medium  whatever,  subjected  to  a  uni- 
form pressure  in  all  directions,  which  lead  to  various  remarkable 
conclusions.     Such  of  these  as  are  independent  of  Joule's  prin- 
ciple (expressed  by  equation  (2)  of  g  SJO),  being  also  indepen- 
dent of  the  truth  or  falseness  of  Carnot's  contrary  assumption 
regarding  the  permanence  of  heat,  are  common  to  his  theory 
and  to  the  dynamical  theory  ;  and  some  of  the  most  important 
of  them*  have  been  given  by  Carnot  himself,  and  other  writers 
who  adopted  his  principles  and  mode  of  reasoning   without 
modification.     Other  remarkable  conclusions  on  the  .-a me  MI!>- 

/  I/          /  V 

ject  might  have  been  drawn  from  the  equation  '        -  '-j-  —  o. 

expressing  Carnot's  assumption  (of  the  truth  of  which  experi- 
mental tests  might  have  been  thus  suggested);  hut  1  am  n»t 
aware  that  any  conclusion  deducible  from  it,  not  included  in 
Carnot's  expression  for  the  motive  power  of  heat  through  finite 
ranges  of  temperature,  has  yet  been  actually  obtained  and  pub- 
lished. 

45.  The  recent  writings  of  Rankine  and  Clausing  contain 
some  of  the  consequences  of  the  fundamental  principle  of  the 
'lyiiiiinical  theory  (expressed  in  the  first  fundamental  proposi- 
tion above)  regarding  physical  properties  of  various  substances ; 
among  which  may  be  mentioned  especially  a  \<  t \  remarkable 
discovery  regarding  the  specific  heat  of  saturated  strain  (in- 
vestigated also  in  this  paper  in  g  58  below),  made  independent  ly 

•  See  aboTr 

in 


THE    SECOND    LAW    OF    THERMODYNAMICS 

by  the  two  authors,  and  a  property  of  water  at  its  freezing- 
point,  deduced  from  the  corresponding  investigation  regarding 
ice  and  water  under  pressure  by  Clausius  ;  according  to  which 
he  finds  that,  for  each  ^°  Cent,  that  the  solidifying  point  of 
water  is  lowered  by  pressure,  its  latent  heat,  which  under  at- 
mospheric pressure  is  79,  is  diminished  by  .081.  The  investi- 
gations of  both  these  writers  involve  fundamentally  various 
hypotheses  which  may  be  or  may  not  be  found  by  experiment 
to  be  approximately  true  ;  and  which  render  it  difficult  to 
gather  from  their  writings  what  part  of  their  conclusions,  es- 
pecially with  reference  to  air  and  gases,  depend  merely  on  the 
necessary  principles  of  the  dynamical  theory. 

46.  In  the  remainder  of  this  paper,  the  two  fundamental 
propositions,  expressed  by  the  equations 

£-£=&         »-•« 

and  ..     , 

Jf=l.f,  (8)  of  g  SI 

are  applied  to  establish  properties  of  the  specific  heats  of  any 
substance  whatever  ;  and  then  special  conclusions  are  deduced 
for  the  case  of  a  fluid  following  strictly  the  "  gaseous  laws  "  of 
density,  and  for  the  case  of  a  medium  consisting  of  parts  in 
different  states  at  the  same  temperature,  as  water  and  saturated 
steam,  or  ice  and  water. 

47.  In  the  first  place  it  may  be  remarked,  that  by  the  defi- 
nition of  J/and  N  in  §  20,  JVmust  be  what  is  commonly  called 
the  " specific  heat  at  constant  volume"  of  the  substance,  pro- 
vided the  quantity  of  the  medium  be  the  standard  quantity 
adopted  for  specific  heats,  which,  in  all  that  follows,  I  shall 
take  as  the  unit  of  weight.     Hence  the  fundamental  equation 
of  the  dynamical  theory,  (2)  of  §  20,  expresses  a  relation  be- 
tween this  specific  heat  and  the  quantities  for  the  particular 
substance  denoted  by  Jfand  p.     If  we  eliminate  M  f rom  this 
equation,  by  means  of  equation  (3)  of  §  21,  derived  from  the 
expression  of  the  second  fundamental  principle  of  the  theory  cf 
the  motive  power  of  heat,  we  find 


dN  _     \fidt  1      ]_dp 


<u> 

137 


M  K  M  0  I  R  S    OX 

which  expresses  a  relation  between  the  variation  in  the  specific 
heat  at  constant  volume,  of  any  substance,  produced  l>y  an  al- 
teration of  its  volume  at  a  constant  temperature,  and  tin-  vari- 
ation of  its  pressure  with  its  temperature  when  the  volume 
is  constant;  involving  a  function,  »,  of  the  temperature,  which 
is  the  same  for  all  substances. 

48.  Again,  let  K  denote  the  specific  heat  of  the  substance 
under  constant  pressure.  Then,  if  dv  and  dt  he  so  related  that 
the  pressure  of  the  medium,  when  its  volume  and  temperature 
are  v  +  dr  and  /  -|-  dt  respectively,  is  the  same  as  when  they  are 
r  and  t  —  that  is,  if 


we  have 

Kdt  =  M<lr  +  Ml. 

Hence  we  find 

_^ 

jf=-±(A'-T).  (i;,) 

dt 

which  merely  shows  the  meaning  in  terms  of  the  two  specific 
heats,  of  what  I  have  denoted  by  M.  Using  in  this  for  .17  its 
value  given  by  (3)  of  §  21,  we  find 

'1'1 


an  expression  for  the  difference  between  the  two  specific  heats, 
derived  without  hypothesis  from  the  second  fundamental  prin- 
ciple of  the  theory  of  the  motive  power  of  In  at. 

r.i.  These  results  maybe  put  into  forms  more  convenient  for 
ii-'-.  in  applications  to  liquid  and  solid  media,  by  introducing 
the  notation: 

tfe 


where  «.-  will  he  the  reciprocal  of  the  compressibility,  and 
coefficient  of  expansion  with  heat. 


THE   SECOND   LAW   OF   THERMODYNAMICS 

Equations  (14),  (16),  and  (3)  thus  become 


K-N  =  v—,  (19) 

M  =  -.Ke;  (20) 

the  third  of  these  equations  being  annexed  to  show  explicitly 
the  quantity  of  heat  developed  by  the  compression  of  the  sub- 
stance kept  at  a  constant  temperature.  Lastly,  if  0  denote  the 
rise  in  temperature  produced  by  a  compression  from  v  +  dv  to 
v  before  any  heat  is  emitted,  we  have 

0  =  ^-r.  -.<&>  =  -_,—  -  srf».  (21) 

N   fi  nK  —  VKe* 

50.  The  first  of  these  expressions  for  0  shows  that,  when  the 
substance  contracts  as  its  temperature  rises  (as  is  the  case,  for 
instance,  with  water  between  its  freezing-point  and  its  point  of 
maximum  density),  its  temperature  would  become  lowered  by  a 
sudden  compression.  The  second,  which  shows  in  terms  of  its 
compressibility  and  expansibility  exactly  how  much  the  tem- 
perature of  any  substance  is  altered  by  an  infinitely  small  alter- 
ation of  its  volume,  leads  to  the  approximate  expression 


if,  as  is  probably  the  case,  for  all  known  solids  and  liquids,  e  be 
so  small  that  B.VKB  is  very  small  compared  with  pK. 

51.  If,  now,  we  suppose  the  substance  to  be  a  gas,  and  intro- 
duce the  hypothesis  that  its  density  is  strictly  subject  to  the 
"gaseous  laws,"  we  should  have,  by  Boyle  and  Mariotte's  law 
of  compression, 

d/=-l,  (22) 

dv          v 

and  by  Dal  ton  and  Gay-Lussac's  law  of  expansion, 
dv        Ev    . 

di^r+m' 

from  which  we  deduce 


MKMOIRS    ON 
Equation  (14)  will  consequently  become 

r/.V  _  d  {„(!  +  A7)~7 

~fo-          ~7t 

a  result  peculiar  to  the  dynamical  theory  and  equation  (1C), 

which  agrees  with  the  result  of  §  53  of  my  former  paper. 

If  V  be  taken  to  denote  the  volume  of  the  gas  at  the  tem- 
perature 0°  under  unity  of  pressure,  (25)  becomes 


52.  All  the  conclusions  obtained  by  Clausius,  with  reference 
to  air  or  gases,  are  obtained  immediately  from  these  equations 
by  taking 


which  will  make  —j-  =  0,  and  by  assuming,  as  he  does,  that  N, 

thus  found  to  be  independent  of  the  density  of  the  gas,  is  also 
independent  of  its  temperature. 

53.  As  a  last  application  of  the  two  fundamental  equations 
of  the  theory,  let  the  medium  with  reference  to  which  M  and 
\  are  defined  consist  of  a  weight  1  —a;  of  a  certain  substance 
in  one  state,  and  a  weight  x  in  another  state  at  the  same  trm- 
perature,  containing  more  latent  heat.     To  avoid  circumlocu- 
tion and  to  fix  the  ideas,  in  what  follows  we  may  suppose  the 
former  state  to  be  liquid  and  the  latter  gaseous  ;  but  tin-  in- 
vestigation, as  will  be  seen,  is  equally  applicable  to  the  case  of 
ft  solid  in  contact  with  the  same  substance  in  the  liquid  or 
gaseous  form. 

54.  The  volume  and  temperature  of  the  whole  medium  !•<•- 
ing,  as  before,  denoted  respectively  by  /•  and  /.  we  shall  have 


if  X  and  y  be  the  volumes  of  unity  of  weight  of  the  substance 
in  tlu>  liquid  and  the  gaseous  states  respectively:  ami  /-.  tin- 
pressure,  may  be  considered  as  a  function  of  /,  depending  solely 
on  the  nature  of  the  substance.  To  express  M  and  .V  for  this 
mixed  medium,  let  L  denote  the  latent  heat  of  a  unit  of  weight  of 
140  • 


THE    SECOND    LAW    OF    THERMODYNAMICS 

the  vapor,  c  the  specific  heat  of  the  liquid,  and  li  the  specific  heat 
of  the  vapor  when  kept  in  a  state  of  saturation.    We  shall  have 

•  Mdv  =  L-r-dv, 
Ndt  =  c(l-x)dt  +  lixdt  +  LC^dt. 


Now,  by  (27),  we  have  7 

(y-\)~  =  l,  (28) 

dv 

and  (y-X)~  +  (l-s)^  +  *||  =  0.  (29) 

Hence  M  =  — ^,  (30) 


N  =  c(l-x)  +  hx-L, 


y-\ 

55.  The  expression  of  the  second  fundamental  proposition  in 
this  case  becomes,  consequently, 


/»  = I-  (32) 


which  agrees  with  Carnot's  original  result,  and  is  the  formula 
that  has  been  used  (referred  to  above  in  §  31)  for  determining 
p.  by  means  of  Eegnault's  observations  on  steam. 

56.  To  express  the  conclusion  derivable  from  the  first  funda- 
mental proposition,  we  have,  by  differentiating  the  preceding 
expressions  for  M  and  N  with  reference  to  t  and  v  respectively, 

^__1_   ^_        L         d(y-\] 
dv   ~-\    dt          —  \'       dt 

_ 
dN     (..  di~didx 


-      (    y  _  \  (y  _  \) 

Hence  equation  (2)  of  §  20  becomes 
dL 


Jdt 


MEMOIRS    ON 

Combining  this  with  the  conclusion  (32)  derived  from  tho  sec- 
ond fundamental  proposition,  we  obtain 

£+<-»=£•  <"> 

The  former  of  these  equations  agrees  precisely  with  one 
which  was  first  given  by  Clausius,  and  the  preceding  in 
gation  is  substantially  the  same  as  the  investigation  by  which 
he  arrived  at  it.  The  second  differs  from  another  given  by 
Clausius  only  in  not  implying  any  hypothesis  as  to  the  form  of 
Carnot's  function  ^i. 

:>',.   If  we  suppose  p  and  L  to  be  known  for  any  temperature. 

equation  (32)  enables  us  to  determine  the  value  of  -f-  for  that 
temperature  ;  and  thence  deducing  a  value  of  dt,  we  have 

dt  =  *^<I/:  (85) 

pL 

which  shows  the  effect  of  pressure  in  altering  the  "boiling- 
point"  if  the  mixed  medium  be  a  liquid  and  its  vapor,  or  tho 
melting-point  if  it  bo  a  solid  in  contact  with  the  >ame  sub- 
stance in  the  liquid  state.  This  agrees  with  the  conclusion  ar- 
rived lit  by  my  elder  brother  in  his  " Theoretical  Investigation 
of  the  Effect  of  Pressure  in  Lowering  the  Pressing-point  of 
Water."  His  result,  obtained  by  taking  as  the  value  for  /«  that 
derived  from  Table  I.  of  my  former  paper  for  the  temperature 
0°,  is  that  the  freezing-point  is  lowered  by  .0075°  Cent,  by  an 
additional  atmosphere  of  pressure.  Clausius,  with  the  other 
data  the  same,  obtains  .oo^:53  as  the  lowering  of  temperature 
by  the  same  additional  pressure,  which  differs  from  my  brother's 
result  only  from  having  been  calculated  from  a  formula  which 

J0 
implies  the  hypothetical  expression  J      '      for  /i.     It  was  by 

applying  equation  (33)  to  determine  -tr  for  the  same  case  that 

ClansiiH  arrived  at  the  curious  result  regarding  the  latent  heat 
of  water  muler  pressure  mentioned  above  (i 

58.  Lastly,  it  may  be  remarked  that  every  quantity  which 
appears  in  equation  (33),  exeept  //.  is  known  with  tolerable  ac- 
curacy for  saturated  steam  through  a  wide  ranije  of  tempera- 
ture; and  we  may  therefore  use  this  equation  to  liu.l  //.  which 
has  never  yet  been  made  an  object  of  experimental  research. 
142 


THE    SECOND    LAW    OF    THERMODYNAMICS 
Thus  we  have  ^         /z 


For  the  value  of  y  the  best  data  regarding  the  density  of  sat- 
urated steam  that  can  be  had  must  be  taken.  If  for  different 
temperatures  we  use  the  same  values  for  the  density  of  saturated 
steam  (calculated  according  to  the  gaseous  laws,  and  Regnault's 
observed  pressure  from  y^,  taken  as  the  density  at  100°),  the 
values  obtained  for  the  first  term  of  the  second  member  of  the 
preceding  equation  are  the  same  as  if  we  take  the  form 


derived  from  (34),  and  use  the  values  of  p  shown  in  Table  I.  of 
my  former  paper.  The  values  of  —  h  in  the  second  column  in 
the  following  table  have  been  so  calculated,  with,  besides,  the 
following  data  afforded  by  Regnault  from  his  observations  on 
the  total  heat  of  steam,  and  the  specific  heat  of  water 


L  =  006.5  +  .305/  -  (.00002/5  +  .OOOOOOO-  • 
The  values  of  —h  shown  in  the  third  column  are  those  derived 
by  Clausius  from  an  equation  which  is  the  same  as  what  (34) 

ri 

would  become  if  /-  --  =-  were  substituted  for  /u. 
1  4-  &t 


t. 

-  A  nccordingto  Table 
I.  of  •'  Account  of 
Carnot's  Theory  " 

-ft  according  to 
Clausius 

0 

1.863 

1.916 

50 

1.479 

1.465 

100 

1.174 

1.133 

150 

0.951 

0.879 

200 

0.780 

0676 

59.  From  these  results  it  appears,  that  through  the  whole 
range  of  temperatures  at  which  observations  have  been  made, 
the  value  of  h  is  negative  ;  and,  therefore,  if  a  quantity  of  sat- 
urated vapor  be  compressed  in  a  vessel  containing  no  liquid 
water,  heat  must  be  continuously  abstracted  from  it  in  order 
that  it  may  remain  saturated  as  its  temperature  rises  ;  and  con- 
versely, if  a  quantity  of  saturated  vapor  be  allowed  to  expand 
143 


MEMOIRS    ON 

in  a  closed  vessel,  heat  must  be  supplied  to  it  to  prevent  any 
part  of  it  from  becoming  condensed  into  the  liquid  form  as  the 
temperature  of  the  whole  sinks.  This  very  remarkable  conclu- 
sion was  first  announced  by  Mr.  Runkine,  in  his  paper  com- 
municated to  this  Society  on  the  4th  of  February  last  year.  It 
was  discovered  independently  by  Clausing,  and  published  in 
his  paper  in  Poggendorff's  Annalen  in  the  months  of  April  and 
May  of  the  same  year. 

60.  It  might  appear  at  first  sight,  that  the  well-known  fact 
that  steam  rushing  from  a  high  -pressure  boiler  through  a  small 
orifice  into  the  open  air  does  not  scald  a  hand  exposed  to  it,  is 
inconsistent  with  ^,he  proposition,  that  steam  expanding  from 
a  state  of  saturation  must  have  heat  given  to  it  to  prevent  any 
part  from  becoming  condensed ;  since  the  steam  would  scald 
the  hand  unless  it  were  dry,  and  consequently  above  the  boil- 
ing-point in  temperature.     The  explanation  of  this  apparent 
difficulty,  given  in  a  letter  which  I  wrote  to  Mr.  Joule  last  Oc- 
tober, and  which  has  since  been  published  in  the  Philfisnjihintl 
Magazine,  is,  that  the  steam  in  rushing  through  the  orifice  pro- 
duces mechanical  effect  which  is  immediately  wasted  in  fluid 
friction,  and  consequently  reconverted  into  heat ;  so  that  the 
issuing  steam  at  the  atmospheric  pressure  would  have  to  part 
with  as  much  heat  to  convert  it  into  water  at  the  teni pi-rat im- 
100°  as  it  would  have  had  to  part  with  to  have  been  condensed 
at  the  high  pressure  and  then  cooled  down  to  loo  .  which  for 
a  pound  of  steam  initially  saturated  at  the  temperature  /  is,  by 
Regnault's  modification  of  Watt's  law,  .305  (t  —  100°)  more  heat 
than  a  pound  of  saturated  steam  at    100°  would  have  to  part 
with  to  be  reduced  to  the  same  state ;  and  the  issuing  steam 
must  therefore  be  above  100°  in  temperature,  and  dry. 

PAW  IV 

On  a  Method  of  discovering  txpfrimfntnllij  t1i><  AV//f//»;/  /«///•»./< 

the  Mechanical  Work  spent  untl  the  JI«if  ]»•<></ nrctl  by 

the  Compression  of  a  Gaseous  Flit  i<l.  * 

61.  The  important  researches  of  Joule  on  the  thermal  cir- 
cumstances connected  with  the  expansion  and  compression  of 

•Prom  Hie  Trantaction*  of  the  Royal  Society  of  Edinburgh,  vol.  we.,  part  2, 
April  17.  1851. 

144 


THE    SECOND    LAW    OF    THERMODYNAMICS 

air,  and  the  admirable  reasoning  upon  them  expressed  in  his 
paper  *  "  On  the  Changes  of  Temperature  produced  by  the  Rare- 
faction and  Condensation  of  Air,"  especially  the  way  in  which  he 
takes  into  account  any  mechanical  effect  that  may  be  externally 
produced,  or  internally  lost,  in  fluid  friction,  have  introduced  an 
entirely  new  method  of  treating  questions  regarding  the  phys- 
ical properties  of  fluids.  The  object  of  the  present  paper  is  to 
show  how,  by  the  use  of  this  new  method,  in  connection  with  the 
principles  explained  in  my  preceding  paper,  a  complete  theoret- 
ical view  may  be  obtained  of  the  phenomena  experimented  on 
by  Joule  ;  and  to  point  out  some  of  the  objects  to  be  attained  by 
a  continuation  and  extension  of  his  experimental  researches. 

62.  The  Appendix  to  my  "Account  of  Carnot's  Theory  "f 
contains  a  theoretical  investigation  of  the  heat  developed  by 
the  compression  of  any  fluid  fulfilling  the  lawsj  of  Boyle  and 
Mariotte  and  of  Dalton  and  Gay-Lussac.  It  lias  since  been 
shown  that  that  investigation  requires  no  modification  when. 
the  dynamical  theory  is  adopted,  and  therefore  the  formula 
obtained  as  the  result  may  be  regarded  as  being  established 
for  a  fluid  of  the  kind  assumed,  independently  of  any  hypothe- 
sis whatever.  We  may  obtain  a  corresponding  formula  applica- 
ble to  a  fluid  not  fulfilling  the  gaseous  laws  of  density,  or  to  a 
solid  pressed  uniformly  on  all  sides,  in  the  following  manner  : 

(53.  Let  Mdv  be  the  quantity  of  heat  absorbed  by  a  body  kept 
at  a  constant  temperature  t,  when  its  volume  is  increased  from 
v  to  v  +  dv  ;  let  p  be  the  uniform  pressure  which  it  experiences 
from  without,  when  its  volume  is  v  and  its  temperature  t  ;  and 

let  p  -f  -j-dt  denote  the  value  p  would  acquire  if  the  temper- 

ature were  raised  to  t  -f  dt,  the  volume  remaining  unchanged. 
Then,  by  equation  (3)  of  §  21  of  my  former  paper,  derived 
from  Clausius's  extension  of  Carnot's  theory,  we  have 


*  Philosophical  Magazine,  May,  1845,  vol.  xxvi.,  p.  369. 

f  Transactions,  vol.  xvi.,  part  5. 

j  To  avoid  circumlocution,  these  laws  will,  in  what  follows,  be  called 
simply  the  gaseous  laws,  or  the  gaseous  laws  of  density. 

§  Throughout  this  paper,  formulae  which  involve  no  hypothesis  what- 
ever are  marked  with  italic  letters  ;  formulae  which  involve  Boyle's  and 
Dulton's  laws  are  marked  with  Arabic  numerals  ;  and  formulae  involving, 
besides,  Mayer's  hypothesis,  are  marked  with  Roman  numerals. 
K  145 


MEMOIRS    ON 

where  p  denotes  C&rnot's  function,  the  same  for  all  substances 
at  the  same  temperature. 

Now  let  the  substance  expand  from  any  volume  Fto  I".  :m  •!. 
being  kept  constantly  at  the  temperature  /.  let  it  absorb  a 
quantity,  //,  of  heat.  *  Then 


"  'fr 


But  if  H'  denote  the  mechanical  work  which  the  substance 
does  in  expanding,  we  have 


and  therefore 

,r      IrfW 

*-tw 

This  formula,  established  without  any  assumption  admitting 
of  doubt,  expresses  the  relation  between  the  heat  developed  by 
the  compression  of  any  substance  whatever,  and  the  mechanical 
work  which  is  required  to  effect  the  compn-ssion.  as  far  as  it 
can  be  determined  without  hypothesis  by  purely  theoretical 
considerations. 

C4.  The  preceding  formula  leads  to  that  which  1  formerly 
gave  for  the  case  of  fluids  subject  to  the  gaseous  laws;  since 
for  such  we  have  ;//.  ,/v.o(1  +  /;/)f  (1) 

from  which  we  deduce,  by  (c), 

ir=;i0r0(l  +  Ay)logy. 

I  1  1  '  J7t  1  • 


log  F  =  j^-^ll:  (a) 


and 

and  therefore,  by  (d), 


which  agrees  with  equation  (11)  of  g  40  of  the  former  paper. 

<;."».  llt-nce  we  conclude,  that  the  heat  rv.>l\v<l  by  any  lluid 
fulfilling  the  gaseous  laws  is  proportional  to  the  work  spent  in 
compressing  it  at  any  given  constant  temperature:  Inn  that 
the  quantity  of  work  required  to  produce  a  unit  of  heat  is  not 
constant  for  all  temperatures,  unless  Carnot's  function  for  dif- 
ferent temperatures  vary  inversely  as  1  4-  AY;  and  that  it  is  not 
the  simple  mechanical  equivalent  of  the  heat,  as  it  was  nnwar- 
140 


THE    SECOND    LAW    OF    THERMODYNAMICS 

mutably*  assumed  by  Mayer  to  be,  unless  this  function  have 
precisely  the  expression  ^ 


This  formula  was  suggested  to  me  by  Mr.  Joule,  in  a  letter 
dated  December  9,  1848,  as  probably  a  true  expression  for  ^, 
being  required  to  reconcile  the  expression  derived  from  Car- 
not's  theory  (which  I  had  communicated  to  him)  for  the  heat 
evolved  in  terms  of  the  work  spent  in  the  compression  of  a  gas, 
with  the  hypothesis  that  the  latter  of  these  is  exactly  the  me- 
chanical equivalent  of  the  former,  which  he  had  adopted  in 
consequence  of  its  being,  at  least  approximately,  verified  by  his 
own  experiments.  This,  which  will  be  called  Mayer's  hypothe- 
sis, from  its  having  been  first  assumed  by  Mayer,  is  also  assumed 
by  Clausius  without  any  reason  from  experiment  ;  and  an  expres- 
sion for  yu  the  same  as  the  preceding,  is  consequently  adopted  by 
him  as  the  foundation  of  his  mathematical  deductions  from 
elementary  reasoning  regarding  the  motive  power  of  heat.  The 
preceding  formulas  show,  that  if  it  be  true  at  a  particular  tem- 
perature for  any  one  fluid  fulfilling  the  gaseous  laws,  it  must 
be  true  for  every  such  fluid  at  the  same  temperature. 

[The  remaining  section*  are  omitted.  They  deal  with  tlte  experimental 
verification  of  Mayer's  hypotJiesis,  with  the  mecfuinwal  energy  of  n  fluid,  and 
trith  the  applications  of  thermodynamics  to  electrical  plienomena\\ 


BIOGRAPHICAL    SKETCH 

WILLIAM  THOMSON,  now  Lord  Kelvin,  was  born  at  Belfast 
in  June,  1824.  His  father,  James  Thomson,  an  eminent  math- 
ematician and  student  of  science,  removed  in  1832  to  Glasgow, 
where  he  occupied  a  position  as  professor  in  the  university. 
His  son  studied  under  him  at  Glasgow  and  at  St.  Peter's  Col- 
lege, Cambridge,  and  was  graduated  in  1845  at  Cambridge,  as 
Second  Wrangler  and  Smith's  Piizeman.  He  was  called  to  the 

*  In  violation  of  Carnot's  important  principle,  that  thermal  agency  and 
mechanical  effect,  or  mechanical  agency  and  thermal  effect,  cannot  be  re- 
garded in  the  simple  relation  of  cause  and  effect,  when  any  other  effect, 
sucli  as  the  alteration  of  the  density  of  a  body,  is  finally  concerned. 
147 


THK    SECOND    LAW    OF   THERMODYNAMICS 

Chair  of  Natural  1'hilnsophy  at  (Jlasirow  University  in 
and  ho  has  since  been  connected  with  that  university.  On  tin- 
completion  of  the  Atlantic  Cable  in  I860,  to  the  success  of 
which  Thomson  had  materially  contributed  by  his  study  of  the 
theoretical  questions  involved  and  by  his  numerous  inventions, 
he  was  knighted,  and  on  New-Year's  day.  lv.i\!.  he  was  raised 
to  the  peerage  as  Lord  Kelvin. 

His  contributions  to  physics  range  over  the  whole  domain  of 
the  science.  The  most  important  of  these  relate  to  electricity 
and  magnetism,  and  to  the  mechanical  theory  of  heat,  of 
great  value  also  are  the  improvements  made  by  him  in  electri- 
cal measurements,  by  his  invention  of  the  mirror  galvanometer 
and  of  the  various  electrometers  which  bear  his  name.  Mis 
work  is  characterized  by  its  great  versatility  and  by  a  peculiarly 
happy  combination  of  profound  theoretical  knowledge  and 
power  of  analysis  with  the  ability  to  invent  and  execute  impor- 
tant experimental  investigations. 


BIBLIOGRAPHY 
BOOKS  OP  REFERENCE 

Clausius.  Die  Mechanische  Warmetheorie. 

Maxwell.  Theory  of  Heat. 

Tail.  Thermodynamics. 

Verdet.  Tlie&rie  Mecanique  de  la  Chaleur. 

Rank  inc.  The  Steam-engine. 

Briot.  La  Chaleur. 

Poincare.  Thennodynamique. 

Planck.  Tliermodyamik. 

Muck.  Die  Principien  der  Wdrmclehre. 

Tail.  Recent  Advances  in  Physical  Science. 


ARTICLES 
On  the  Validity  of  the  Second  Law 

Rankine.  Phil.  Mag.  (IV.),  4,  p.  358. 

Holtzmaiin.  Pogg.  Ann.,  82,  p.  445. 

Decker.  Dingler's  Pol.  Jour.,  148,  pp.  1,  81,  161,  241. 

Him.  Cosmos,  22,  pp.  283,  413.     (Also  in  Exposition  Analytique  et 

Experimental  de  la  Theorie  Mecaniqite  dc  la  Chaleur.) 

Wand.  Carl's  Repertorium.  4,  p.  281,  369. 

Tail.  Phil.  Mag.  (IV.),  43. 

The  foregoing  articles  contain  criticisms  of  Clausius's  form  of  tke  Second 
Law.  They  are  answered  by  Clansius  in  Appendices  to  kis  book,  Die 
Mechanische  WdrmetJieorie,  2d  ed. ,  187U. 


On  Mechanical  Analogies  to  the  Second  Law 

Clausius.         Pogg.  Ann.,  93,  p.  481. 

Phil.  Mag.  (IV.),  35,  p.  405. 

Pogg.  Ann.,  142,  p.  433. 

Ibid.,  146,  p.  585. 

Rankine.         Phil.  Mag.  (IV.),  30,  p.  241. 
149     , 


THE   SECOND    LAW    OF   THERMODYNAMICS 

Boltzmann.     Per.  der  Wiener  Akatl.,  58,  II.,  p.  188,  195. 

I  bid..  68.  II.,  pp.  :.-V  71-j. 

Ibid.,  76,  II.,  p. 

Ibid.,  78.  II.,  pp.  1,  738. 

Szily.  Pogg.  Ann  .  14-1,  pp.  -J9o,  302  ;  149,  p.  74 

Burbury.         Phil.  Mag.  (V.),  1,  p.  61. 

Of  these  the  later  papers  by  Bolt/.munn  and  that  by  But  bury  <: 
the  relation  between  the  Second  Law  and  the  Theory  of  Probability.     The 
same  subject  is  developed  by  Boltzmann   in  his  book,  I'vrUsungen  uber 
Gattlieorie,  2d  part. 


INDEX 


Caloric,  Action  of,  7  ;  Conservation 

of,  Assmmd  by  Carnot,  20. 
Carnot,  Biographical  Sketch,  60. 
Carnot's  Cycle,  11,  18.  73,  78. 
Carnot's  Function,  91,  93,  123,  128, 

135. 
Carnot's  Principle,  13,  20  :  Modified 

by    Clausius,    90;    Modified     by 

Thomson.  115,  117. 
Clausius,  Biographical  Sketch,  107. 
Cycle.     See  Carnot's  Cycle. 


E 

Efficiency  of  Engines,  in  Terms  of 
Carnot's  Function,  36,  126  ;  Com- 
pared, 42,  58. 

Engine,  Ideal,  11,  18,  74,  114. 


Gases,  Heat  Relations  of  (Carnot), 
23,  25,  28,  31,  38  ;  (Clausius),  71, 
77,  84,  85,  86,  87,  88,  91  ;  (Thom- 
son), 139,  146. 


11 


Hent.  See  Caloric.  Effects  of,  3; 
Total  and  Latent,  69  ;  Mechanical 
Equivalent  of,  105.  127. 

Heat,  Relations  of,  to  Work  ;  (Car- 
not), 8  ;  (Cbiusius),  66,  68  ;  (Thom- 
son), 111,  115,122,  126,133. 


Liquids  and  Vapors.     See  Vapors. 


Motive  Power,  Carnot's  Definition 
of,  6  ;  Produced  by  Suitable  Trans- 
fer of  Heal,  7,  9/10  ;  Relation  of, 
to  Difference  of  Temperature,  36; 
Produced  by  Several  Agents  Com- 
pared, 41. 

S 

Second  Law  of  Thermodynamics 
(Clausius),90, 118  ;  (Thomson),116. 

Steam-Engine,  3,  7,  50. 

Substances,  Heat  Relations  of,  137, 
138,  139,  146. 

T 

Thomson  (Lord  Kelvin),  Biograph- 
ical Sketch,  147. 


Vapors,  Heat  Relations  of  (Carnot), 
43  ;  (Clausius).  78.  81,  92,  96,  105  ; 
(Thomson),  140,  142. 

w 

Water  Vapor,  Departure  of,  from 
Mariotte's  and  Gay-Lussac's  Laws, 
95 ;  Negative  Specific  Heat  of, 
105,  143. 

Work.  See  Motive  Power.  Equiva- 
lent to  Heat.  66,  115  ;  Relation  ol, 
to  Unit  of  Heat,  105,  127. 


151 


Univer 

Sou 

Lit 


